ROBERT  E.  PETERSON'S  CHEAP  EdIaVIONAL  ^m\lS. 


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BY     ENOCH     LEWIS. 


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ROBERT  E.  PETERSON'S  CHEAP  EDUCATIONAL  SERIES. 


TREATISE  ON  ALGEBRA, 


CONTAINING  THE 


MOST  USEFUL  PARTS  OF  THAT  SCIENCE, 


ILLUSTRATED    BY 


A  COPIOUS  COLLECTION  OF  EXAMPLES: 


DESIGNED  FOE  THE  USB  OF  SCHOOLS  AND  COLLEGES. 


BY    ENOCH    LEWIS. 


FOURTH  EDITION,   REVISED  AND  ENLARGED. 


PHILADELPHIA  : 
PUBLISHED  BY  ROBERT  E.  PETERSON, 

N.      \V.  C0RNP:K  of  fifth  and  ARCH  STREETS. 


1852. 


*^ 


Entered  according  to  Act  of  Congress  in  the  year  1852, 

BY    ROBERT    E.    PETERSON, 

In  the  Clerk's  Office  of  the  District  Court  of  the  Eastern 

District  of  Pennsylvania 


Deacon  &  Peterson,  Printers, 
66  South  Third  Street. 


PEEFACE  TO  THE  FOURTH  EDITION. 


Since  the  first  edition  of  this  work  was  issued,  a  number 
of  years  have  elapsed,  during  which  several  treatises  on 
this  branch  of  science  have  been  given  to  the  public  5  some 
of  which  are  evidently  the  productions  of  able  hands ;  still 
the  reasons  which  induced  the  author  of  The  Practical  Ana- 
lyst to  offer  his  work  to  public  acceptance,  appear  applicable 
to  the  present  day.  The  generality  of  pupils  who  undertake 
to  acquire  a  knowledge  of  algebra,  require  a  manual  which 
includes  the  general  principles,  and  processes  of  the  science 
without  needless  expansion,  or  very  abstruse  speculations. 
The  work  has  been  carefully  revised,  and  such  changes  in- 
troduced as  were  judged  needful  to  adapt  it  to  the  present 
improved  state  of  the  science.  A  number  of  examples,  not 
contained  in  the  former  editions,  have  been  inserted  in  this, 
and  the  original  object  of  smoothing  the  path  of  the  pupil 
and  diminishing  the  labor  of  the  instructor,  has  been  kept 
steadily  in  view.  Though  the  design  of  this  treatise  is 
rather  to  meet  the  wants  of  students  in  our  schools,  than 
to  supply  that  class  who  may  incline  to  dive  into  the  most 
refined  intricacies  of  the  analytic  art ,  yet,  even  to  the 
latter  class,  this  tract  will  probably  be  found  a  safe  and 
correct  introduction. 

3 


l^30B114 


PREFACE. 


The  mathematical  sciences  have  been  considered,  since 
the  early  periods  of  their  existence,  among  the  noblest  ob- 
jects of  human  inquiry.  Of  these,  next  to  arithmetic,  geo- 
metry occupies  the  first  place,  in  regard  both  to  importance 
and  to  time.  The  Grecian  philosophers  cultivated  this 
branch  of  knowledge,  with  an  ardor  and  industry,  which 
manifest  their  high  opinion  of  its  value.  In  the  Ionian 
and  Pythagorean  schools,  geometry  was  considered  an  in- 
dispensable preliminary  to  the  study  of  philosophy.  When 
a  person,  ignorant  of  geometry,  applied  for  his  instructions 
to  the  philosopher  Xenocrates,  he  is  said  to  have  made  this 
laconic  reply :  "  Thou  hast  not  the  handles  of  philosophy." 

The  perspicuity  of  geometrical  reasoning,  the  accurate 
and  inimitable  dependence  of  its  arguments,  and  the  un- 
faltering certainty  of  its  conclusions,  are  eminently  calcu- 
lated to  form  the  mind  to  habits  of  attention,  and  to  a 
regular  and  forcible  cbncatenation  of  its  ideas.  No  won- 
der, then,  that  it  was  so  highly  prized  by  that  acute  and 
inquisitive  people. 

Two  modes  of  procedure,  according  to  the  different  ob- 
jects in  view,  were  adopted  by  the  ancient  geometers: 
synthesis  and  analysis. 

Synthesis,  or  composition,  consists  in  the  direct  solution 
of  a  problem ;  or,  the  demonstration  of  a  proposition  by 
a  series  of  arguments,  regularly  deduced,  from  self-evident 
truths,  or  from  other  propositions  previously  established. 
This  method  is  very  proper  for  conveying,  with  clearness, 
to  another,  those  truths  which  are  completely  understood 
by  the  instructor.  The  works  of  ihe  ancient  geometers, 
which  have  escaped  the  ravages  of  the  barbarous  ages,  are 
mostly   synthetical.      The  Elements  of  Euclid,  so  well 

5 


VI  PREFACE. 

known  in  our  modern  schools  of  geometry,  furnish,  proba- 
bly, the  most  complete  specimen  of  the  ancient  synthesis 
extant.  And  it  is  a  remarkable  fact  in  the  history  of 
science,  that  the  geometry  of  Euclid,  though  written  two 
thousand  years  ago,  and  some  time  anterior  to  any  other 
mathematical  tract  that  has  reached  us,  is  still  one  of  the 
best  elementary  works  on  that  subject  which  has  ever  ap- 
peared. 

Analysis,  or  decomposition,  on  the  other  hand,  assumes, 
as  known,  the  proposition  which  is  to  be  examined,  or, 
as  already  effected,  the  solution  which  is  to  be  made,  and 
thence  proceeds  to  examine  the  consequences  necessarily 
resulting  from  such  supposition,  until,  in  case  of  a  theorem, 
a  conclusion  is  attained,  the  truth  or  falsehood  of  which  is 
already  known,  whence  the  correctness  or  absurdity  of  the 
supposition  becomes  established ;  or,  in  case  of  a  problem, 
such  relations  are  determined  as  prove  the  possibility  or 
impossibility  of  the  solution.  In  synthesis,  observes  Mon- 
tucla,  we  proceed  from  the  known  to  the  unknown,  from 
the  trunk  to  the  branches  5  in  analysis,  we  proceed  from 
the  unknown  to  the  known,  from  the  branches  to  the 
trunk. 

The  analytical  method  is  frequently  indispensable,  when 
new  problems  are  to  be  solved,  or  new  theorems  investi- 
gated. No  doubt,  many  propositions  which  the  ancients 
have  transmitted  to  us  in  the  synthetical  form,  owe  their 
discoveries  to  analysis. 

Of  this  method  of  procedure,  the  mathematical  collec- 
tions of  Pappus^  and  the  work,  De  Sectiones  Rationis^  of 
Apollonius,  furnish  the  principal  specimens  which  the 
ancient  geometers  have  left  us. 

In  the  ancient  geometry,  the  magnitudes  under  consi- 
deration, were  mostly  presented  to  the  mind  through  the 
medium  of  representations,  as  similar  a^practicable  to  their 
antitypes.  In  some  instances,  however,  this  analogy  was 
entirely  abandoned,  as  »in  the  fifth  book  of  Euclid's  Ele- 
ments, where  right  lines  are  the  only  representatives  used, 
yet  the  reasoning  is  so  conducted,  as  to  be  equally  applica- 
ble to  magnitudes  of  every  kind,  and  even  to  abstract  num- 


PREFACE.  Vn 

bers.  In  this  instance,  we  may  discover  a  commencement 
of  that  species  of  generalization,  which  forms  so  conspicu- 
ous a  feature  in  our  modern  mathematics. 

T.he  ancient  analysis  furnished  the  germ  of  that  branch 
of  mathematics,  which,  in  the  hands  of  the  moderns,  has 
become  the  great  master  key  to  all  the  rest;  but  it  does 
not  appear  to  have  assumed  the  character  of  a  distinct 
science,  till  after  the  commencement  of  the  Christian  era. 

The  earliest  writer,  in  whose  works  the  science  of  algebra 
is  distinctly  seen,  was  Diophantus,  a  mathematician  of  the 
Alexandrian  school.  The  time  in  which  he  lived  is  not 
precisely  known,  but  it  was  not  later  than  the  fourth  cen- 
tury, as  the  daughter  of  Theon,  the  amiable  and  accom- 
plished Hypathia,  who  died  about  the  beginning  of  the 
fifth,  wrote  a  commentary  on  his  w^orks. 

Whether  Diophantus  was  the  inventor,  or  only  an  im- 
prover of  algebra,  cannot  now  be  known ;  the  latter  sup- 
position, however,  is  the  more  probable,  as  the  science  in 
his  hands  exhibits  a  degree  of  maturity,  which  it  can  hardly 
be  supposed  to  have  attained  in  the  first  period  of  its  ex- 
istence. 

A  part  only  of  the  original  work  of  Diophantus  is  now 
to  be  found ;  in  this  he  does  not  explain  the  first  principles 
of  the  science,  but  teaches  the  solution  of  a  great  variety 
of  difficult  questions,  in  that  branch  of  the  subject,  now 
called,  from  him,  the  Diophantine  Algebra,  or  the  indetermi- 
nate analysis^  applied  to  equations  of  the  higher  orders. 
The  various  questions,  if  original,  which  he  has  formed, 
and  the  address  with  which  he  has  conducted  their  solu- 
tions, necessarily  inspire  his  readers  with  a  high  opinion 
of  his  invention  and  discernment.  His  work,  written  in 
the  original  Greek,  was  discovered  in  the  Vatican  Library, 
about  the  middle  of  the  sixteenth  century. 

Though  the  science  of  algebra  appears  to  have  originated 
among  the  Greeks  of  the  Alexandrian  school,  the  inhabi- 
tants of  western  Europe  derived  their  knowledge  of  it  from 
the  Arabs,  who  are  by  some  supposed  to  have  been  its  in- 
ventors. 

Dr.  Wallis  observes,  that   they    differ  essentially  from 


via  PREFACE. 

Diophantus  in  their  manner  of  expressing  the  powers. 
The  Greek  analyst  calls  the  2d,  3d,  4th,  5th,  6th,  etc. 
powers,  the  square,  cube,  squared  square,  squared  cube, 
cubed  cube,  etc. ;  each  power  being  designated  by  the  two 
inferior  powers  of  which  it  is  the  product.  But  the  Ara- 
bian algebraists  denominate  the  5th  power  the  first  sur- 
solid,  the  6th  the  squared  cube ;  being  the  square  of  the 
cube,  and  not  the  product  of  a  square  and  cube ;  the  7th 
the  second  sursolid,  and  so  of  the  other  powers.  Hence, 
he  infers,  that  the  Greek  and  Arabian  analyses  were  not 
derived  from  a  common  source. 

With  due  regard  for  the  opinion  of  this  eminent  scholar, 
it  appears  quite  as  rational  to  suppose,  that  the  Arabian 
mathematicians  may  have  borrowed  from  their  tutors,  the 
Greeks,  this  branch,  with  the  rest  of  the  mutilated  sciences, 
and  adopted,  in  the  denomination  of  their  powers,  a  mode 
of  expression  peculiar  to  themselves,  as  to  believe,  that 
while  the  other  parts  of  Grecian  science  were  sought  with 
avidity,  this  was  permitted  to  sleep  amidst  the  dust  of  ne- 
glected libraries,  till  the  same  thing  had  been  reinvented 
by  a  people  much  less  advanced  in  scientific  knowledge, 
and  less  remarkable  for  invention  than  their  predecessors.* 

Whether  the  science  of  Algebra  was  invented  by  the 
Arabs,  or  borrowed  from  the  Greeks,  the  name  is  unques- 
tionably of  Arabic  origin.  The  names  given  by  the  Arabs, 
for  they  used  a  plurality,  were  algebra  valmucahala.  These 
words,  according  to  Lucas  de  Burgo,  signify  restauratio  et 
opposition  restoration  or  rebuilding,  and  opposition.  Golius 
defines  the  word  gebera,  or  giabera,  by  religavit  consolidavit, 
it  bound  or  consolidated ;  and  mocabulat,  by  comparatio 
opposition  comparison,  opposition. 

*  In  the  progress  of  science,  there  always  appears  a  regular  depend- 
ence in  the  successive  stages.  The  discoveries  of  one  age  are  the  re- 
sults of  those  made  in  the  former.  Logarithms  were  invented,  when 
the  discoveries  in  astronomy  and  trigonometry  had  rendered  their  use 
indispensable.  The  discoveries  of  Newton  could  not  have  been  made, 
even  by  that  gigantic  genius,  in  the  time  of  Copernicus;  nor  could 
Columbus  have  led  his  trembling  companions  across  the  Atlantic,  be- 
fore the  invention  of  the  mariner's  compass. 


PREFACE.  IX 

By  these  words  they  probably  designed  to  indicate  the 
general  objects  of  the  science.  The  quantity  whose  value 
is  sought,  is  commonly  interwoven  with,  or  hound  to  other 
quantities,  in  such  a  manner  as  to  form  one  or  more  equa- 
tions or  comparisons  of  quantities  set  in  opposition  to  each 
other.  These  equations  are  then  transformed,  or  rebuilt^ 
till  the  unknown  quantity  is  brought  out  in  opposition  to  a 
given  or  known  one.^ 

The  most  ancient  authors  on  algebra  among  the  Arabs, 
are  Mohammed  ben  Musa,  and  Thebit  ben  Corah.  The 
former  is  described  by  Cardan  as  the  inventor  of  the  me- 
thod of  solving  equations  of  the  second  degree,  a  discovery 
in  which  he  was  certainly  anticipated  by  the  mathemati- 
cians of  the  Alexandrian  school.  From  the  title  given  to 
his  book,  it  has  been  inferred,  that  he  flourished  during  the 
reign  of  Almamon,  or  in  the  early  part  of  the  ninth  century. 

Whether  the  Arabian  algebraists  proceeded  beyond  the 
solution  of  equations  of  the  second  degree,  is  an  unsettled 
question.  An  accurate  knowledge  of  the  mathematical 
sciences  is  seldom  combined,  in  the  same  individual,  with 
an  extensive  acquaintance  with  Arabic  literature,  and, 
therefore,  little  is  certainly  known  on  this  subject.  The 
Bodleian  Library  in  England,  and  that  of  the  Escurial  in 
Spain,  are  said  to  possess  a  great  number  of  Arabic  works 
on  the  subject  of  algebra.    • 

Leonard  of  Pisa,  who  lived  near  the  beginning  of  the 
thirteenth  century,  impelled  by  a  thirst  for  mathematical 
knowledge,  traveled  into  Arabia  and  other  parts  of  the  east, 
and  on  his  return,  first  communicated  to  his  countrymen 
the  science  of  algebra.  I  do  not  find  that  any  of  his  writ- 
ings on  that  subject  have  ever  been  published. 

In  a  treatise  upon  trigonometry,  by  Regiomontanus,  of 
Franconia,  written  about  the  year  1464,  some  problems 
are  solved  by  algebra,  in  which  he  refers  to  the  rules,  a? 
though  commonly  known. 


*The  name  almncahala  was  adopted  by  some  of  the  Italian  writers, 
and  it  is  thus  designated  in  some  of  the  works  of  Cardan ;  but  the 
term  algebra  appears  now  irrevocably  fixed  upon  it. 


X  PREFACE. 

But  the  earliest  European  author,  whose  works,  specially 
on  this  subject,  have  been  published,  was  Lucas  de  Burgo, 
before  mentioned.  He  was  a  Franciscan,  who  traveled  in 
the  east,  either  in  pursuit  of  knowledge,  or  for  some  pur- 
pose not  now  known,  and  after  his  return,  taught  mathe- 
matics at  Naples,  Venice,  and  Milan*.  His  work,  in  which 
the  rules  of  algebra  are  laid  down,  was  first  printed  in  1494. 
In  this  the  science  appears  very  far  below  our  modern 
algebra.  The  rules  for  the  solution  of  adfected  quadratic 
equations,  are  given  in  semi-barbarous  Latin  verse,  and  the 
different  cases  separately  treated.  His  solutions  do  not 
rise  to  equations  above  the  second  degree. 

The  solution  of  cubic  equations  appears  to  have  been 
first  effected  by  Scipio  Ferrei,  professor  of  mathematics  at 
Bologna,  about  the  beginning  of  the  sixteenth  century. 

This  solution,  however,  included  but  one  case,  namely, 
that  in  which  the  first  and  third  powers  only,  of  the  un- 
known quantity,  were  involved.  Some  questions,  including 
this  case,  being  afterward  proposed  to  Nicholas  Tartaglia, 
an  eminent  mathematician  of  Brescia,  he  discovered  a 
general  solution  of  equations  of  the  third  degree.  This 
discovery  he  communicated  to  Jerome  Cardan,  a  physician 
of  Milan,  under  an  injunction  of  secrecy.  Cardan,  not- 
withstanding, having  found  the  demonstration,  published 
it  in  his  work,  De  Arte  Magna,  in  1545.  This  is  now 
commonly  called  the  method  of  Cardan. 

Cardan  first  remarked  the  plurality  of  roots  in  quadratic 
equations,  and  their  distinction  into  positive  and  negative. 
The  solution  of  equations  of  the  fourth  degree  was  accom- 
plished soon  after  the  discovery  of  Tartaglia,  by  Lewis 
Ferrari,  a  pupil  and  coadjutor  of  Cardan. 

Some  intricacies  belonging  to  equations  of  the  third  de- 
gree were  further  unraveled  by  Raphael  Bombelli,  of 
Bologna,  and  published  in  1579. 

The  algebraists,  whose  labors  have  been  noted,  expressed 
known  quantities  by  their  proper  numerical  characters,  and 
therefore  their  solutions  were  destitute  of  that  generality 
which  constitutes  so  prominent  a  feature  of  our  present 
analysis.     Their  modes  of  solution  were  applicable  to  all 


PREFACE.  XI 

similar  problems,  but  their  results  were  confined  to  par- 
ticular questions.  Francis  Vieta,*  by  using  letters  of  the 
alphabet  to  represent  known  &.s  well  as  unknown  quanti- 
ties, gave  an  extent  and  generality  to  the  science  which  it 
did  not  possess  before.  By  this  change,  algebraists  were 
enabled  to  include,  in  a  single  solution,  a  whole  class  of 
problems,  and  to  obtain  the  result  in  each  particular  case, 
by  simple  substitution.  He  taught  the  method  of  taking, 
from  an  equation,  its  second  term,  and  thus  reducing  ad- 
fected  quadratics  to  simple  quadratics  j  and  all  cases  of 
cubic  equations  to  the  case  solved  by  Ferrei.  He  also 
taught  the  solution  of  cubic  equations,  having  three  possi- 
ble roots,  by  the  trisection  of  an  angle.  He  made  numer- 
ous improvements  in  algebra,  and  furnished  the  germs  of 
some  discoveries  which  have  since  grown  up  under  other 
names. 

Thomas  Harriott,  an  English  analyst,  followed  in  the 
steps  of  Vieta,  and  made  several  important  improvements 
in  the  science.  He  first  adopted  the  plan  of  placing  all 
the  terms  of  an  equation  on  one  side  of  the  sign  of  equality, 
and  zero  on  the  other ;  and  showed  that  an  equation  thus 
expressed,  may  be  always  formed  by  the  multiplication  of 
binomial  factors.  This  naturally  led  to  the  discovery,  that 
every  equation  has  as  many  roots,  or  values  of  the  unknown 
quantity,  as  there  are  units  in  the  index  of  its  highest 
power.  Harriott,  notwithstanding  this  observation,  lay 
directly  in  his  road,  does  not  appear  to  have  made  it.  The 
complete  development  of  negative  and  impossible  roots, 
was  left  to  exercise  the  ingenuity  of  succeeding  inquirers. 
Vieta  employed  the  large  letters  of  the  alphabet,  and  in- 
dicated their  powers  by  initials  placed  over  them,  as  ex- 
ponents are  now  used ;  Harriott  substituted  small  letters, 
and  denoted  their  powers  by  repetitions  of  the  letter ;  thus 
instead  of  ./^%  A"^  etc.,  he  wrote  aa^  aaa^  etc.,  a  small 
change,  indeed,  but  still  an  obvious  improvement  in  the 
notation.     His  Artis  AnaSyticse  Praxis,  containing  his  dis- 

*  This  eminent  mathematician  was  born  at  Fontenoy,  in  Poitou,  in 
1540,  and  died  in  1603. 


Xll  PREFACE. 

coveries,  was  published  after  his  death.*"  He  was  born  at 
Oxford,  in  1560,  and  died  in  1621. 

The  celebrated  French  philosopher,  Eene  Descartes, 
contributed  largely  to  the  advancement  of  algebra.  He 
explained  the  nature  of  negative  roots,  and  taught  the  man- 
ner of  finding  their  number  by  the  changes  of  the  signs  in 
the  general  equation.  The  use  of  exponents,  as  now  ap- 
plied, is  attributed  to  him ;  as  is  also  the  method  of  inde- 
terminate co-efficients. 

The  first  application  of  algebra  to  geometry,  was  long 
prior  to  the  time  of  Descartes,  yet  those  sciences  are  in- 
debted to  him  for  that  intimate  union,  which  has  since  con- 
tributed so  extensively  to  the  improvement  of  both.  Des- 
cartes was  born  in  1596,  and  died  in  1650. 

Soon  after  the  time  of  Descartes,  the  analytical  science 
took  a  flight,  which,  if  followed,  would  lead  me  far  beyond 
the  bounds  of  a  preface.  This  sketch  of  the  history  will, 
therefore,  be  closed  with  the  remark,  that  this  science,  as 
enriched  by  the  discoveries  of  Newton,  Leibnitz,  and  others, 
has  become,  in  the  hands  of  our  modern  philosophers,  the 
torch  to  guide  them  through  the  most  intricate  labyrinths 
of  science  ;  that  by  its  light  they  have  traced  the  motions 
of  the  celestial  bodies  through  all  their  mystic  dance,  and 
penetrated  many  of  the  recesses  of  nature,  where,  without 
its  aid,  they  must  have  been  bewildered  and  lost. 

The  following  work  was  undertaken  from  a  persuasion, 
that  the  books  on  algebra,  used  in  our  schools,  were  none 
of  them  entirely  adapted  to  the  wants  of  a  large  class  of 
pupils,  many  of  whom  do  not  enjoy  the  leisure,  or  the 
talents  requisite  for  penetrating  the  depths  of  science,  and 
yet  are  desirous  of  attaining  a  knowledge  of  this  subject, 
sufficient  to  qualify  them  for  studying  successfully  the  com- 
mon practical  branches  of  mathematics,  to  which  this  serves 
as  a  key.  In  the  most  popular  treatise  on  this  science, 
with  which  our  schools  have  been  furnished,  the  progress 
of  the  student  appears  to  me  neecflessly  obstructed,  by  diffi- 

*  Bossut  says  it  was  published  in  1620 ;  Montucla  and  others  say  in 
1C31. 


PREFACE.  XUl 

culties  near  the  commencement,  which,  to  a  common  in- 
tellect, are  almost  insuperable. 

My  object  has  been  to  present  the  most  useful  parts  of 
the  science  in  such  order,  that  no  very  abstruse  process 
should  be  required,  before  the  pupil  had  been  sufficiently 
exercised,  to  acquire  the  requisite  skill.  The  expedients 
demanded  for  solving  the  questions  are  mostly  pointed  out 
before  they  are  called  for  in  practice. 

In  a  popular  treatise  on  a  subject  which  has  engaged  the 
attention  of  so  great  a  number  of  authors,  many  of  them, 
unquestionably,  endued  with  talents  of  the  highest  order, 
it  would  be  idle  to  expect  much  originality.  My  object 
has  been  to  smooth  the  path  of  the  student,  and  diminish 
the  toil  of  the  tutor.  The  work,  with  all  its  imperfections, 
is  submitted  to  the  inspection  of  the  public. 
2 


CONTENTS 


Definitions,         ...       -        -         -         -         Page  15 

Addition, 18 

Subtraction, 21 

Multiplication,    - 22 

Division,    --------  26 

Involution,  or  the  Raising  of  Powers,           -         -  32 

Evolution,  or  the  Extraction  of  Roots,          -         -  36 

Algebraic  Fractions, 41 

Equations, 50 

Simple  Equations, -  51 

Quadratic  Equations, 84- 

Promiscuous  Examples, 101 

Ratios, 106 

Variations  of  Quantities, Ill 

Series,         -         -         -         -         -         -         -         -114 

Summation  of  Series, 128 

Differential  Method,    -         -         -         -        .  129 

Construction  of  Logarithms,        -         -         -         -  133 

Surds, 142 

Equations  in  General, 155 

Indeterminate  Problems, 166 

Miscellaneous  Examples, 176 

xiv 


ALGEBRA. 


DEFINITIONS. 

Article  1.  Algebra,  or  specious  arithmetic,  is  the 
science  of  computing  by  symbols  or  general  characters. 

2.  Quantities,  of  whatever  kind,  are  usually  denoted  by 
letters  of  the  alphabet. 

3.  The  relations  of  quantities,  and  the  operations  to  be 
performed  on  them,  are  indicated  by  the  following  charac- 
ters, thus : 

4.  The  sign  +  plus^  or  more^  indicates  addition,  as 
a-\-b  signifies  that  b  is  added  to  a, 

5.  The  symbol  —  minus^  or  less^  indicates  subtraction ; 
thus,  a — Z>,  signifies  that  b  is  subtracted  from  a. 

The  characters  +  and  —  are  called,  by  way  of  emi- 
nence, the  signs  of  the  quantities  to  which  they  are  pre- 
fixed. 

6.  Multiplication  is  denoted  by  the  sign  X ,  into^  placed 
between  the  factors,  as  axb;  or  by  a  period,  as  a.b;  or, 
more  frequently,  by  joining  the  letters,  like  letters  in  a 
word,  as  ab;  each  of  which  expressions  denotes  the  pro- 
duct of  a  and  b, 

7.  Division  is  indicated  by  the  sign  -r-  %,  placed  be- 
tween the  terms ;  or  by  writing  the  dividend  above,  and 

the  divisor  below,  a  horizontal  line ;  thus  a-H^,  or  -7-  de- 
notes the  quotient  of  a  when  divided  by  b, 

15 


16  DEFINITIONS. 

8.  The  difference  of  two  quantities,  when  it  is  unknown 
which  is  the  greater,  is  indicated  by  the  sign  -v,  placed 
between  them ;  thus,  arj)^  or  b^a^  denotes  the  difference 
of  a  and  b. 

9.  :  :  :  :  indicates  proportion;  thus,  a  :  b  :  :  c  :  d^  may 
be  read,  a  has  to  b  the  same  ratio  that  c  has  to  d, 

10.  =  equal  to,  signifies  that  the  quantities  between 
which  it  is  placed,  are  equal  to  each  other,  thus,  b-\-c 
=  fl — d+e,  is  a  combination,  called  an  equation,  which 
signifies,  that  when  the  operations  indicated  by  the  signs 
are  performed  on  the  quantities  placed  on  each  side  of  the 
sign  of  equality,  the  results  are  equal  to  each  other. 

11.  A  simple  quantity  is  that  which  consists  of  one 
term  only,  viz.  a  quantity  denoted  by  a  single  letter,  or 
several  letters  and  figures,  connected  by  the  sign  of  mul- 
tiplication or  division,  expressed  or  understood,*  as  a,  abc, 

5cd;  J- 

12.  A  compound  quantity  consists  of  two  or  more  sim- 
ple quantities  connected  by  the  signs  of  addition  or  sub- 
traction, as  a — 6,  ab — ac  +  bd, 

13.  A  root  is  a  number  or  quantity,  from  which  a  power 
is  conceived  to  arise. 

14.  A  power  of  a  number  or  root,  is  the  product  of  a 
unit,  multiplied  continually  by  the  given  root,  any  pro- 
posed number  of  times ;  and  the  figure  or  quantity  which 
indicates  the  number  of  multiplications  thus  made,  is  called 
the  index  or  exponent;  thus,  1x5x5x5x5=6255  and, 
IXaX  axaxaxa,  or  aaaaa,  are  the  4th  and  5th  powers 
of  5  and  a  respectively,  and  are  usually  expressed  by  the 
root  with  the  index  of  the  power  set  over  it:  as  5'*,  a^; 
hence  a^=  1,  whatever  value  may  be  assigned  to  a, 

15.  The  second  power  is  called  the  square;  the  third 
power,  the  cube;  the  fourth  power,  the  biquadrate,  etc.  of 
their  respective  roots. 

*  An  absolute  number,  though  containing  numerous  digits,  is  con- 
sidered as  a  simple  quantity. 


DEFINITIONS.  17 

16.  The  radical  sign^  -/,  prefixed  to  a  quantity,  indi- 
cates the  square  rootj  ^,  the  cube  root;  -y,  the  fourth 
root,  etc.     Roots  are  also  expressed  by  fractional  expon- 

1 
ents;  thus,  y/a,  or  a"',  denotes  the  square  root  of  a;   ^/a, 

1  •  .2 

or  a^,  the  cube  root  of  a;  and  v^ri^,  or  a^,  the  cube  root 
of  the  square  of  a, 

17.  A  root  which  cannot  be  accurately  expressed  in 
numbers,  is  called  a  surd^  or  irrational  quantity,  a^  V  5? 

18.  A  quantity  which  has  no  radical  sign,  or  which 
having  a  radical  sign,  admits  of  an  accurate  extraction  of 
the  root  indicated  by  the  sign,  is  called  rational ;  thus  a, 
%/16,  4/Z>^,  are  rational  quantities. 

19.  When  a  compound  quantity  has  a  line  drawn  over 
it,  or  is  enclosed  in  brackets,  the  operation  indicated  by  a 
preceding  or  subsequent  sign,  is  to  be  performed  on  the 
whole  considered  as  a  simple  quantity;  thus,  a-{-b — cXc?, 
and  (b-\'C)x  (d — e),  signify  that  the  compound  quantities 
connected  by  the  sign  X ,  are  multiplied  together ;  and 
y/ab-\-dc,  '^{ab — cd-{-ef,l  signify  the  sauare  root,  and 
the  cube  root  respectively,  of  the  compound  quantities 
which  are  preceded  by  the  radical  signs. 

20.  A  number  prefixed  to  a  letter,  or  combination  of 
letters,  is  called  the  co-efficient;  thus,  in  the  expression 
Sab,  3  is  the  co-efficient. 

21.  A  compound  quantity,  consisting  of  two  terms,  is 
called  a  binomial,  as  a  +  b;  one  of  three  terms,  a  trinomial, 
as  ab-{-ac-{-de. 

22.  Like  quantities  are  those  which  consist  of  the  same 
letters  similarly  involved ;  as  ab,  Sab,  5ab, 

23.  Unlike  quantities  consist  of  different  letters,  or  dif- 
ferent powers  of  the  same  letters,  as  a,  ab,  Sa\  4^ab^. 

24.  A  simple  quantity  is  usually  termed  a  monomial. 

=*So  called  from  radix^  a  root. 


18  ADDITION. 

A  compound  quantity  is  termed  a  binomial,  a  trinomial,  or 
polynomial,  according  as  it  consists  of  two,  three  or  many 
terms.  Thus^  a -\-b,  ab  +  cd,  are  binomials,  a  +  ^-f^j  CLb  + 
cd+ef,  are  trinomials,  etc. 

In  the  solution  of  problems,  it  is  usual  to  denote  known 
or  given  quantities,  by  the  initial  letters,  a,  b,  c,  etc.,  and 
unknown  ones,  by  the  final  letters,  v,  a?,  y,  etc. 

The  following  examples  are  given  for  the  purpose  of 
exercising  the  student  in  the  application  of  the  algebraic 
signs. 

Required  the  numerical  values  of  the  following  combi- 
nations, supposing  a=7,  i=6,  c=5,  c?=3,  e=2. 

a'  +  Uc—3de=:34^3+  120—18=445. 

(a5  +  cc/)x(35c  +  4«6Z)  =  (42+15)X(90  +  84)  =  57xl74 
=  9918. 

a^  +  cd'—3be=:S03. 

b^ + d'd—lce—^bc^  223. 

{ac^'-^^bd)  X  6ac=  159425.  , 

b'  VaH^'-f«5=20 

4 


a'b'hl3cd^ 

-2^^I3S--^^-^^^^ 
6ad^—2ace^ 
~7bT3d~'^^^'^^ 

s/4^a^+ld'+63be=zl3 

a^—a^b  +  3ad^—4<lce6S 

_  __ — -—  o 

ab — cd  -f  ae         34 


Section  I. 

ADDITION. 

Case  I. 

25.   When  the  quantities  are  like,  a^d  have  like  signs. 

Add  all  the  co-efficients  together,  and   to   their  sum 


ADDITION.  19 

annex  the  given  literal  expression,  prefixing  the  common 
sign.* 


EXAMPLES. 

3a 

2be 

2a'b 

—     3xy^ 

2a— 3b 

5a 

3be 

3a'b 

—       a??/^ 

5a     6b 

la 

be 

Wb 

—    40??/- 

la— 6b 

4a 

5be 

a'b 

—     Ixy'' 

8a— 46 

6a 

Sbe 

ba'b 

—     9xy^ 

9a     4^b 

a 

9be 

a'b 

—  llxy'' 

Ida—  b 

26a 


5x^y — 76c   +   ab  6xy^ — lac  4fab^  +  3a^c —  bc^ 

3a73y— 66c  _|_4a5  ^xy; — 3ac  bab'' -}- 4<a''c —  bc^ 

bx^y^3bc  +   ab  4a:y^ — 5ac  3a¥  +  6a^c — 2bc^ 

Ix^y—  i6c4-4a6  3a??/^— 2ac  8a6^-f  Sa^c— 86c3 

Sx^y — 2i6c  +  3a6  8xy^ —  ac  4^ab^-\-3a^c — Ibc^ 

3x^y—lbc  -\-6ab  bxyl—lac  ab''-\-6a^c—3b(^ 

4,x^y — 36c  +2a6  3xy^ — 2ac  5a63  +  5a2c — 46c3 

5a?3y — |5c  +3a6  4<xy^ — 13ac  2a63+   a^c — 56c3 


Case  2. 
26.  When  the  quantities  are  alike,  but  have  unlike  signs. 

Take  the  difference  between  the  sum  of  the  positive  and 
the  sum  of  the  negative  co-efficients,  prefix  the  sign  of  the 
greater,  and  annex  the  literal  part. 

=*  The  sign  +  or  — >  is  frequently  prefixed  to  quantities  which 
are  not  preceded  by  others,  to  or  from  which  they  are  required  to  be 
added  or  subtracted  ;  a  preceding  quantity,  however,  may  always  be 
supposed.  Algebraic  quantities  are  said  to  be  positive,  or  negative, 
according  as  they  are  preceded  by  the  signs  -f  or  — ;  the  former 
being  always  understood  where  none  is  expressed. 


20  ADDITION 


2ab^-\-  6cd^  — Sx^  — Sa^^  +  4a?y 

-5ab^  +  4^cd^  6x'^y^  4<a-b  +  2x^y^ 

ab^ — 7cd^  7a? Y  -        — Sa^b-\-   x^y^ 

-Sab^  +  bcd^  — 4fX^y^  la^b — 3x^y^ 

Qab^ —  cd''  — 9xY  —  a^b  +  ^x^ 


6ab   ■i-4^cd^ 

—Sx^ 

Sab^—5c^d^ 

—  5ab^  +  2c^d' 

-  a¥  +  4<c^d'^ 
lab'^—Sc^d^ 
8ab^—5c^d^ 

—eab^  +  lc^-d^ 

S^ay 

— 9\/ay 

— ly/ay 

3  s/ ay 

by/ ay 

— ^y/  ay 

^Oax —  Iby/xy 

— 31ax-\-   bby/xy 

21  ax —  6by/xy 

18ax —  bby/xy 

—  4aa?+  13b  s/  xy 

—3Qax-\-llby/xy 

Case  3. 

27.   When  there  are  unlike  quantities,  and  different  signs. 

Collect  the  like  quantities  as  in  the  foregoing  cases,  and 
connect  the  results  by  their  proper  signs. 

EXAMPLES. 

4.aa:— 130   +   3x^  lax  3a''—'7bc+6ad 

5a?2  +  3ax  +   9x^         — 5xy  ^xy—lad — 5a» 

Ixy — 4^x^    -h   90  — 3ax  8ad—6xy — 35c 

6x^ — 5s/x-}-3xy  2xy  bxy-\-2a^ — 4ac? 

40  — 5ax  — lOxy         — 3ab  4^bc  +  3ad —  xy 

2ax + 20x^  —6x^  4<ax-3xy-3ab 


SUBTRACTION.  21 

6x^y —  lax — 3bc  14<ax —  20?^ 

26c  +   3x^y-{-4<ax  6ax-\-   3xy 

Sax —  56c   -\-lad^  Sy^ —  4^ax 

2bc  +   3ad^—6ax  3x^+26 

x'^y —  56c  +4<ax  ^xy —  by^ 

Sad^ —  3ax — 4<x^y  bx'^ —  Sax 

ax —ISad^'—Sbc  '7y^—26 


x^y^  +  4^ad^  — 76  V  ax'^y^ +3abY — bc^x'^y^  +   b^x^ 

Wc^—lad^  +2ac*  3¥xY — 4^aa?y  +  4a'crj^3  — 5a  6y 

3a(^  +66V  +5a?y  2a¥y'  —3b^xY—2axY  +2cV/ 

Sad^—5ac^  —66  V  4^c^xY  +2ao(^y^—3¥xY  +4<abY 

bxhf+   ad^ —  6V  2b^oifyz  +  4<abY  +2axY  — 2c^xY 

353^2  — 5 -j.'?y2 — 2ac*  Sc^x^  — 2axY  +  bb^x^yz  +  9^62^3 

QxY — 2ac*  — 6ad^  ^ab^  — 3c^xY —  «^Y  — Ib^x^yz 

1.  Add  3«  +  6,  la — 56,  76 — 3a,  4a — 26,  and  8a — 56 
together.  Kesult,  19a — 46. 

2.  Add  5aa?  +  76c— 3a2,  86c  +  4a3— 2aa7,  lla^ — 46c— 6a-r 
and  2aa? — 36c  +  6a2  into  one  sum. 

Result,  86c+18a2 — ax. 

3.  Add  together  the  following  quantities,  labc  +  ba^d 
—3xy%  2aH-\-bxy^—3abc,  Uc^-^-Sxy^-^-Sa^  Uc^—Sabc— 
a%  and  a^d — lOxy^ — a6c — 76c^ 

Kesult,  10a3j+66c^ 

SUBTRACTION. 

28.  Change  the  signs  of  the  quantities  which  are  to  be 
subtracted,  or  conceive  them  to  be  changed,  then  collect 
the  terms  as  in  addition. 

EXAMPLES. 

From  8xy—3+   6x —  y         3x^ — 2a6        5a62 — 7^20+   ad 
Take  3xy — 7 —  6x+by  x^ — 7a6        3a62 — 2¥c—4^ad 


5a??/+4+ 1207— 6y  ^x>/^'- 


22 


MULTIPLICATION. 


From  17a5— 3a?y  +  552 
Take  llab  +  4^xy  +  '7b^ 


Sax — 55c 
lax — 76c 


9aey+4<bc 
2aey-{-lbc 


i: 


-yi 


From  l[x''—2a^x—18  +  6bc 
Take  3x^—1  a  y/x—  9— 36c 


4fad — Ixy  +  Sbc 
bad —  y'^z  +  6bc 


-  -^ 


1.  From  a  +  b  take  a — b.  Result,  26. 

2.  From  8a — 12a?  take  5a  +  3x.  Result,  3a — 16x. 

3.  From  12a-\-10b-{-13ax — 3ab,  take  la — 5b  +  3ax, 

Result,  ba+lbb+10ax—3ab, 

4.  From   the   sum   of  3a6 — lax   and   lab-\-3ax,    take 
/^ab — 3ax — 4<xy,  Result,  6ab — ax-{-4<xy, 

5.  From  5a?^ — 4a?2/  +  5,  subtract  4<x''^ — ^xy+9. 

Result,  x^ — 4. 

6.  From  ax^ — bx^+x  subtract  po?^ — cx^+ex. 

Result,  (a—p)  x''+(c—b)x^+(l—e)x. 

1,  From  bx^  -\-  cx'^ — dx + e  take  px" — ga?^-|-ra? — 5. 

Result,  (b—p)x^ +(c  +  q)x^'^  (d+r)x+e+ s. 


MULTIPLICATION. 

Case  I. 
29.  When  the  factors  are  both  simple  quantities. 

Multiply  the  co-efficients  together,  and  annex  all  the 
letters  to  the  product.* 

*From  Def.  14,  it  is  obvious,  that  the  product  of  two  or  more 
powers  of  any  letter,  may  be  expressed  by  that  letter  with  an  index 
equal  to  the  sum  of  the  indices  of  the  factors ;  thus,  a^a^=aaa,aa=a5» 


MULTIPLICATION.  23  • 

J^ote, — When  the  signs  of  the  factors  are  like,  the  pro- 
duct must  be  made  positive  i  when  unlike,  negative. 

EXAMPLES. 

lOah  2xy^  Sla^b  bad 

ba^c  — Qx'^y  Sab^c  Sa^b^o 

bOa^bc    — 12a?y 

— acx^  Sxyz         ba^bc         — ^ad^  boo^y'^ 

— bc^x         — 2x^y         lac^  — ba^b         — 4<x^y^ 


Case  2. 

30.   When  one  of  the  factors  is  a  compound  quantity. 

Multiply  the  simple  factor  into  each  term  of  the  com- 
pound ones,  and  connect    the   products   by   their   proper 


a^ 

examples. 

xy+2y^                 2ab—2b^ 
Sax                    — 4<¥ 

a*—2a^b+a%^ 

xy^+4bbc^—9 
bcx 

Sx^y + bay^—lbc^          Sc^+ax 
— 2a%                            bax 

Case  3. 
31.  When  both  factors  are  compound  quantities. 

Multiply  the  whole  multiplicand  by  every  term  of  the 
multiplier,  and  collect  the  several  products  as  in  addi- 
tion. 


24, 


4j 

a+b 
a+b 

MULTIPLICATION. 
EXAMPLES. 

a^  +  ab  +  b^ 
a^—ab  +  b^ 

a^+ab 
ab+b^ 

a^-\-a^b  +  a^b^ 
— a^b — a^b^ — ab^      - 
a'^b^+ab^  +  b^ 

a^+2ab+b'- 

a*          ^a^b^         +6* 

a+b            x^+xy+y'^           x^+x^y  +  x^y^+xy^  +  y* 
a — b            X  — y                   X  — y 

2x^+3xy—4t^y^ 
3x — 5y 

2ab — 3ac+4fbc           x^ — ^y-\-y^ 
ab-\-  ac —  be           x — y 

1.  Multiply  ba^  +  ^b^  by  ba^—^^bK 

Result,  25a4— 166*. 

2.  Multiply  a?3  +  3x^y  -{•3xy^+ y^  by  x^  -r  2xy  +  y^. 

Eesiilt,  x^-\-5x^y-\-10x^y^+  lOx^y^-i-  bxy'^+y^ 

3.  Multiply  3x^+2xy+6y^  by  x^ — ocy+y^ 

Result,  3a7* — x^y-{-6x^y^ — 3xy^  +  5y*. 
4«.  Multiply  6x^+10xy  +  6y^  by  5x^ — 10xy+5y^ 

Result,  25a?* — 50x^y^+25y\ 

5.  Multiply  a^  +  3a^b+3ab^+¥  by  a^—3a'b  +  3ab^—b\ 

Result,  a^ — 3a''b^+3a^b^ — ¥. 

6.  Multiply  a^ — ab+¥ — be  by  a'^+ab—b^+bc. 

Result,  a^—a-b^ + 2ab^—2a¥c + 2¥c—b^c^—¥. 

7.  Multiply  0?* — x^y+x^y^ — ^y'^+y'^  by  x^+xy+y\ 

Result,  x^+oc*y^ — x'^y'^+x^y*+yK 


MULTIPLICATION.  25 

8.  Multiply  x+y  by  x — 2/,  and  the  product  hyx'^+y^, 

Kesult,  X* — y*. 

9.  Find  the  continued  product  of  a^ — 2ab+h%  a"^-\-2ab 
+5^,  a^-{.a"b-{-ab'^'{-b^  and  a — b. 

Result,  a^—2a^b''  +  2a''b^—b^ 

10.  Find  the  continued  product  of  3a?+6,  3x+2y  3x — 2, 
and  3x—6,  Kesult,  Slx^—SQOx'^+lU. 

11.  What  is  the  continued  product  of  3a — 26,  4a — 3b, 
4<a  +  3b,  and  3a  +  '2b?  Ans.,  lUa^—Uda^b^  +  36b\ 

12.  Multiply  a^—a^b  +  a"b'' — a'^b^+ab* — b^-,  by  a+b. 

Result,  a«— S'^. 

13.  Multiply  x'^-^-x'^y+xy^i-y^j  by  o?^ — x^y+xy^ — y^. 

Result  0?^ + x^y^ — x^y* — y^. 

When  the  factors  consist  of  several  terms  with  the 
powers  regularly  ascending,  the  multiplication  may  be 
effected  by  the  co-efficients  alone,  and  the  literal  part  be 
afterwards  supplied.  Thus  in  the  5th  example,  we  may 
take 

1+3+3+1 
l_3  +  3— 1 

1+3+3+1 
—3—9—9—3 

3+9+9+3 
_1_3_3— 1 

1  0—3  0+3  0—1.  Then  to  supply  the  literal 
part,  we  observe,  that  a^,  a^=a^,  the  first  term;  and  as  the 
powers  of  a  descend  and  those  of  b  ascend  in  a  regular 
order,  the  terms,  without  their  co-efficients,  must  be,  a^,  a^6, 
a*Z)%  a^b^,  a^b\  ab\  b\  But  the  2d,  4th  and  6th  co-efficients 
are  each  =0,  therefore  the  terms  are  a^ — 3a^b^ -\'3a'^b^ — b^. 

In  like  manner,  the  7th  example  may  be  performed 
thus : 

3 


26  DIVISION. 

1—1+1-1+1 
1+1+1 


1—1+1^1+^ 

1_14-1._1  +  1 
1^1+1—1  +  1 

1  0  +  1—1  +  1  0  +  1.  In  which  the  2d  and  6th  co- 
efficients are  =0,  hence  the  2d  and  6th  terms  x^y  and  xy^^ 
do  not  appear  in  the  product. 


DIVISION. 

Case  I.* 

32.  When  the  divisor  and  dividend  are  both  simple 
quantities. 

Divide  the  co-efficient  of  the  dividend  by  that  of  the 
divisor;  expunge  from  the  dividend  such  letters  as  are 
common  to  it  and  the  divisor,  when  they  have  the  same 
exponents ;  when  the  exponents  are  not  the  same,  subtract 
the  exponent  of  the  divisor  from  that  of  the  dividend,  and 
use  the  remainder  as  the  index  of  the  letter  in  the  quotient ; 
write  in  the  quotient  the  letters  which  have  not  been  ex- 
punged, with  the  co-efficient  above  determined. 

JVb^e. — In  division,  as  in  multiplication,  like  signs  in  the 
divisor  and  dividend  require  the  positive  sign  in  the  quotient  j 
unlike  signs,  the  negative. 

*  If  the  student,  who  is  just  entering  into  this  science,  should 
express  the  powers  by  repetitions  of  the  letters,  the  process  will  be 
more  simple,  and  the  grounds  of  the  method  indicated  by  the  rule  be 
rendered  obvious. 


3ab)9a^bcd 
Sacd 


DIVISION. 
EXAMPLES. 

'5x^y^)15x*y^z^ 


—3x^z^ 


27 


2ad)6a^d^c 


llab^pla^^ 


3a^c)  12a*bc^         la^b)—%  la^bc^ 


— 3^0:2:) — ISa^xy^z^ 


lbabc^)—4<5a^b^c'^ 


33.   When  the  dividend  is  not  a  multiple  of  the  divisor. 

Set  them  down  as  a  vulgar  fraction,  the  divisor  being 
the  denominator;  and  divide  the  terms  by  such  numbers 
and  quantities  as  are  common  to  both ;  the  result  will  be 
the  fractional  answer. 


Iba^bc     Sac 
10aZ>2~26 

ISaxy^    — 9y 
—Sax^y~    4j7^ 

1. 

Divide  2Sxy^  by  7a? Y- 

Result,  -^ 

2. 

Divide — lOabc  by  15abd 

Eesult,     3^ 

3. 

Divide  limn  by  22m^ 

Kesult,  2^, 

4. 

Divide  19xyz  by  2bx^yz^ 

Case  2. 

^^^""'25^ 

34.   PFAe/i  ^Ae  dividend  is  a  compound  quantity,  and  the 
divisor  a  simple  one. 

Divide  every  term  of  the  dividend  by  the  divisor,  as  in 


28  DIVISION. 

the  former  case,  and  connect  the  results  by  their  proper 
signs, 

EXAMPLES. 


3a 


-z=z6x''—5ay  +  9a^ 


2.  Divide  15axy^ — SOb'^xy — lOx'^y^  by  6xy. 

3.  Divide  30a^c— ISac^+lS^c  by  Sac, 

4.  Divide  15a^bc — 15acx'^-{-5ad^  hj — 5ac, 

5.  Divide  21a^x'^ — la^x^ — 14<ax  by  lax. 

6.  Divide  72a2Z>Y— 12a%+36a%  ^7  ^^%- 

7.  Divide  2Sx^y^-\-3bxY — 1(oc^y^z  by  7a?y. 

^  Case  3. 

35.    When   the  divisor  and  dividend  are   compound 
quantities. 

Arrange  the  terms,  both  of  the  divisor  and  dividend,  in 
such  manner  that  the  higher  powers  of  some  one  letter 
shall  always  precede  the  lower;  then  find  the  first  term  of 
the  quotient,  by  using  the  first  of  the  divisor  and  dividend, 
as  in  the  first  case.  Multiply  the  whole  divisor  by  the 
quotient  thus  found;  subtract  the  product  from  the  divi- 
dend, as  in  common  arithmetic;  and  proceed  till  the  divi- 
dend is  exhausted.  The  remainder,  if  any  thing  remain, 
with  the  divisor  for  a  denominator,  must  be  annexed,  with 
its  proper  sign,  to  the  quotient, 

EXAMPLES. 

a+b)a^  +  3a^b  +  3ab^+b\a^  +  2ab+b'' 
a3+  a^b 


'Ha-b  +  Sab^ 
2a''b  +  2ab^ 


ab^+b^ 
a¥-\-b^ 


DIVISION.  29 

^^' 

— ^a^b  -f  9a^Z>2 — lOa'^b^  a^^2ab  +  6^ 


3a3^3 — la^b^-^bab* 
3^353 — 6a2^>3^3fij54 

— a^b^-i-^ab^ —  b^ 
—265 

3.  Divide  a?* — 4-a7^y  +  6a?y — 4a?y3  4-y*  by  x — y  ^ 

Eesult,  x^ — 3x-y  +  3xy^ —  y, 

4^.  Divide  a^ — b^  by  a — b.  Result,  a  +  b, 

5.  Divide  a^  +  b^  by  a+b.  Result,  a'^-r-ab+b^ 

6.  Divide  y^^3y^—4^y  + 12  by  y^—4<.     Result,  y—3, 

7.  Divide  b^—3b^x''+3b^x^—x^  by  ¥—3b^x  +  3bx''—x\ 

Result,  53 + 36^0?  +  3bx'' + a?^. 

8.  Divide  24a*— 6*  by  3a— 65. 

(^03^4 

Result,  8a^+  16a^b  +  32ab'^+6W+\ ^ 

oa — K)o 

9.  Divide  a*-r4^a^b'''—32b^hj  a  +  2b. 

Result,  a^—2a^b  +  Sab''—16b^ 

10.  Divide  oj^+y^  by  x+y. 

Result,  0?^ — a?*y4-a?^y^ — x^^+xy* — j/^H — ^ — 

11.  Divide  x^ — y^  by  x — y. 

Result,  0?^ + x'^y  +  x^y^  +  x^y^ + a:*2/*  4  x'^y^  4-  a?^^^  4-  a^y^  4  y^. 

12»  Divide  a*  -4-  4a^6 + Ga^^^ + 4a6^  +  6* + ^a^c  4-  12a26c  + 
12ab^c  +  U^c-\'6a''c^-\-12abc^+6b^c''-{-^ac^  +  4<bc^-\-c^  by  a^-f 
2a5+2ac+62+2k-}-c2. 

Result,  a^+2ah  +  2ac+b^+2bc  +  c^. 
o 


30  DIVISION. 


PROMISCUOUS  EXAMPLES. 


1.  Eequired  the  sum  of  5ac-f7^e,  Sac — ibe,  lac — Sbe, 
8ac4-46e,  and  2ac — She,  Ans  26aC'\-be, 

2.  Collect  3ax+5xy — lz%  Sxy — 2ax+4^z%  lax  +  Sz^ — 
xy^  and  6z^ — 4^ax+2xy,  into  one  sum ;  and  subtract 
therefrom  the  sum  of  the  following  quantities,  3ax — 5a:y4- 
4}Z%  Ixy — bax — 62?%  and  bax — 2xy-\-^z'^, 

Result,  ax-\'^xy-\-z^, 

3.  Required  the  product  of  o?^ +37^?/ +  0:2/24-2/3  by  x — y, 

Ans.  0?* — ?/*. 

4.  What  is  the  sum  of  the  products  of  a'^+Sx^y-fSo?^'^ 
+y3  by  07+2/,  and  x'^ — 2xy+y^  by  x'^ — 2xy+y^1 

Ans.  2x'+12xY+2y\ 

5.  Divide  x^ — y^  by  x — y,  and  from  the  quotient  sub- 
tract the  product  of  x^-{-y^  by  a?^ — xy. 

Result,  2x^y  +  2xy^+y\ 

6.  Multiply  3a^—9a^b-{-9ab^—3b^  by  a''+2ab+b%  and 
a^-\.3a''b-\-3ab^.+b^  by  3a''—6ab~\-3b%  and  find  the  sum  of 
the  products.  Result,  6a^—12a^b'^-\'6ab\ 

7.  If  a* — 6*,  be  divided  by  a — 5,  and  the  product  of 
a^+ab  +  b^  by  a — b,  subtracted  from  the  quotient,  what 
will  the  remainder  be  1  Ans.  a'^b-]-a¥-\-2b'^, 

8.  Divide  x^+Sx7y+28x^y''+5t6xY+l[0xY+^Q^Y+ 
2Sx'^y^ + Sxy^ + y^  hvx^  +  2xy +y^:  and  multiply  ar* — 4<x"y 
+6x^y^ — 4<xy^-\'y^  by"a?'^ — 2xy+y^,  and  find  the  sum  and 
difference  of  the  results. 

J.      ,     (  Sum,  2x^-\^30xY+^0xY+2y^ 

'^Difference,  12x^y-{-4<0xY+12xy\    ^ 

9.  Divide  a^— 6^  by  a^-ha^b^  +  b^-,  and  multiply  a""— 
ah+b^  by  a  +  b;  and  find  the  sum  and  difference  of  the 
results.  Sum,  2a\     Diff.  2bK 

When  a  co-efficient,  either  numerical  or  literal,  is  com- 
mon to  every  term  of  the  divisor  or  dividend,  the  process 


DIVISION.  31 

maybe  shortened,  by  first  dividing  by  that  co-efficient,  then 
using  the  quotient  instead  of  the  given  quantity,  and  mul- 
tiplying or  dividing  the  result  by  such  co-efficient,  accord- 
ing as  it  was  found  in  the  dividend  or  divisor. 

13.  Divide  Sax"^ — 9ax^y-{'9axy~ — 3ay^  by  x^ — 2a7^  +  y^ 
Here  the  dividend  =3a  (x^ — 3x^y+3xy^ — y^).     Hence  if 
we  divide  x^ — 3x^y-{-3xy^ — y'^  by  x^ — 2xy+y^,  and  multi- 
ply the   resulting   quotient  x — y  by    3a,  we  shall  have 
3ax — 3ay,  the  true  quotient  required. 

14.  Divide  5a* — 56*  hja^+b\  Result,  5a^—5b\ 

15.  Divide  '7a^—28a^b  +  4<2a^b^—28ab^^7b^  by  3a''— 
6ab+3b^    y  Result,  J(a^—2a6 +62.) 

When  the  dividend  and  divisor  contain  no  more  than 
two  letters,  with  regularly  ascending  and  descending 
powers,  the  operation  may  frequently  be  abridged  by 
omitting  the  letters,  and  proceeding  with  the  co-efficients. 
Thus  in  the  third  example :  page  29. 

-3  +  3 — 1  And  we  perceive 
from  the  first  term 
of  the  dividend  and 
divisor  that  the  first 
term  of  the  quo- 
tient is  a?^5  hence 
the  result  is  readily 
discovered. 


1- 

-1)1—4  +  6—4+1(1. 
1—1 

—3+6 
—3  +  3 

3     4 

,3-3 

—1  +  1 

—1  +  1 

In  the  10th  example,  the  dividend,  when  the  powers 
are  regularly  arranged,  is  x^+0x^y-{-0x^y^-{-0x^y^-{'0x'^y*-\- 
Oxy^ — y^.     Hence  we  may  proceed  in  this  manner. 


92  INVOLUTION. 

1  +  1)1  +  0+0  +  0  +  0-1-0—1(1—1  +  1—1  +  1—1 

1+1 


—1  +  0 
—1—1 


And  as— =a?55weread- 

1  +  0  ^ 

^  J]  J  ily  discover  the  powers  of 

X  and  y  which  must  be 
annexed  to  these  co-effi- 
cients. 


1  +  1 


—1  +  0 
—1—1 


1  +  0 

1+1 

—1—1 
—1—1 


Section  II. 
INVOLUTION,  OR  THE  RAISING  OF  POWERS. 

36.  To  involve  a  given  quantity  to  any  power,* 

Multiply  the  quantity  by  itself  as  many  times  as  there 
are  units  in  the  index  of  the  given  power  diminished  by 
one. 

EXAMPLES. 

1.  Required  the  5th  power  of  a^b^c, 
(a^b^cfz^aH^^cK 

*When  the  quantity  to  be  involved  is  a  simple  one,  multiply  its 
index,  or  indices,  by  that  of  the  power  proposed,  observing  that  the 
even  powers  of  negative  quantities  are  positive,  and  the  odd  ones 
negative. 


INVOLUTION.  33 

2.  Required  the  4th  power  of  2b — c. 

2b  —c 
2b  —c 


U^—2bc 
— 2bc  +c3 

453 — 45c   _^c2 
2b— c 

863—  862c  +  2bc^ 
—  463c  4-  46c2 — c3 

26— c 

166*— 246"c  +126V— 26c3 
— 86'^c  +  1262c2— 66c3+c4 

166*— 326'^c  4-2462C2— 86c3+c* 

3.  Find  the  3d  power  of  ba^, 

4.  Required  the  5th  power  of  — 2a. 

5.  Required  the  4th  power  of  a +07. 

6.  Required  the  square  of  3a — 26. 

7.  Required  the  6th  power  of  5a +6. 

8.  Required  the  3d  power  of  a +26+ c. 

37.  A  binomial  is  raised  to  any  power,  with  great  fa- 
cility, by  the  following  method.* 

Set  down,  for  the  first  term  of  the  power,  the  first  term 
of  the  root  involved  to  the  given  power. 

The  succeeding  terms,  without  the  co-efficients,  consist 
of  the  successive  powers  of  the  first  term  of  the  root,  regu- 
larly descending,  joined  to  the  powers  of  the  second,  regu- 

*  I'he  reasons  for  this  process  are  given  in  a  subsequent  part  of  this 
^^ork.  -See  Art.  98. 


34^  INVOLUTION, 

larly  ascending  from  the  first  ^  the  common  difference  of 
the  indices  being  one. 

The  co-efficient  of  the  second  term  is  the  index  of  the 
given  power;  and  if  a  co-efficient,  already  found,  be  multi- 
plied by  the  exponent  of  the  leading  quantity,  or  first  term 
of  the  root,  contained  in  the  same  term  of  the  power,  and 
divided  by  the  number  of  terms  to  that  place ,-  the  quotient 
will  be  the  co-efficient  belonging  to  the  next  term.* 

JSTote, — To  determine  the  signs  of  the  different  terms, 
it  must  be  remembered  that  the  even  powers  of  negative 
quantities  are  positive,  and  the  odd  powers  negative. 

EXAMPLES. 

Eequired  the  5th  power  oi  a-\-x, 

a^  a^x  a^x"^  a^x^  ax^        x^ 

The  co-efficients,  1,  5,  —^—  =  10, — - — =  10,  etc. 

Hence,  (a-{-xy=a^+6a*X'^  10a^x^-\-  lOa^x^-^  bax^+x^. 

Eequired  the  6th  power  of  2x — 3y, 

(2xy     (2xy.3y     (2x)\(3yy     (2xy,(3yy     (2a?)3.(3y)* 
(2xX3yy     (3yy 

Or,  640^6  96a?5y  14<4<x^y^  216xY  324a?Y  4^S6xy^  729/. 

6x5  15x4 

The  co-efficients  1,  6,--^r-=15,— ^— =20,  15,  6,  1. 

Hence,  (2x—3yy=64<x^—616x^y+2160xY-^4^320x'y^ 
+4860a?2y*— 2916a:/+7292/«. 

2.  Eequired  the  3d  power  of  a+y, 

Ans.  a^  +  3a^y+3ay'^+y^ 

=*  The  co-ellicients  of  any  two  terms,  equally  distant,  the  one  from 
the  beg;inning,  and  the  other  from  the  end,  are  alike ;  hence,  the  com- 
putation of  the  first  half  of  them  will  supply  the  whole. 


INVOLUTION.  35 

3.  Required  the  4th  power  of  a — 5. 

Ans.  a^—^a^h+^a^b^—^ah^^h^ 

4.  Required  the  7th  power  oi  x+y. 

Result,  a?7+7a?sy+21a?y+35a?*y3^,  etc. 

5.  Required  the  8th  power  of  a — c. 

Result,  a«— 8a7c+28«6c2— 56a5cH'70a*c4,  etc. 

6.  Required  the  4th  power  of  2  +  ^. 

Result,  16+32a?+24a?H8^+^- 

7.  Required  the  3d  power  of  a — 3Z>. 

Result,  a^—^a''b-\'Tlah''—2W, 

This  method  of  involution  is  easily  extended  to  trino- 
mials, quadrinomials,  etc.,  by  considering  two  or  more  of 
the  terms  as  a  single  compound  one. 

8.  Required  the  4th  power  of  a-\'h — c. 

Here  considering  b — c  as  the  second  term  of  the  bino- 
mial, we  have  {a-\-b — c)*=a*+4a^(6 — c)+Qa\b — c)^-}- 
4a(6 — c)^+(6 — c)*j  and  involving  b — c,  by  the  same  me- 
thod, to  the  powers  indicated,  we  have  (b — cy=b^ — 26c + 
c2 ;  (6  _  cy = b^—Sb^c  4-  3bc^—c^ ;  (b  —  cy= ¥  —  ^¥c  4- 
6Z>2c3  —  45c^  +  c*  j  consequently,  (a-\-b  —  c)*=  a*  +  ^a^b  — 
4a3c+6a262 — 12a2^c-f  6a2c2+4a63 — l2abH-\-12abc^ — ^ac^J^ 

^4 4,^3^2  _|.  6^2c2 ^J)c3,  _J_  c4, 

9.  Required  the  3d  power  oi  x-\-y-\-z. 

Result  a?3 -)- 3a:2y  ^  ^xy'^^y^  +  307^2: + 6a?yz+ 
3/2;  +  3a:2;2+  33/2:24-03. 

10.  Required  the  second  power  of  a  4-6 — c — d. 

Result,  a2^.2a64-62— 2ac— 26c4-c2— 2a6?— 
^bd+'Zcd-ird^ 

11.  Required  the  4th  power  oi  x-\-y — 32r. 

Result,  a?*4-4a?^2/4-6a?'^2/2_j.4a?y3  4_/ — 12a?'^2: — 
3<6x''yz—3Qxy'-z—11y^z  4-  ^4^x^z^+  lOSxyz^ 
+b4<y^z^—lOSxz^—10Syz^+81z\ 


36  EVOLUTION. 

12.  Required  the  5th  power  of  a+6+c. 

Result,  a^  +  5a45+10a^62+l0«253+5aZ>*+55 

I0a''c^+30a^bc^+30ab^c^+10b^c^+10a^c^ 
+  20abc^+l0b^c^+6ac^  i-bbc^+c\ 

13.  Required  the  3d  power  of  a+b — c — d. 

Result,  a^  +  3a^b  +  3ab^+b^ — 3a''c — 3a^d—6abc 
— Qabd—3b''c—3b^d+  3ac^ + 3bc^ +6acd+ 
6bcd+3ad^+3bd^—c^ — 3c^dr—3cd^—d\ 


EVOLUTION,  OR  THE  EXTRACTION  OF  ROOTS. 

Case  I. 
38.  To  extract  the  root  of  a  simple  quantity. 

Extract  the  root  of  the  co-efficient,  for  the  numeral  part ; 
and  divide  the  index  or  indices  of  the  literal  part,  for  the 
exponents  of  the  root.* 

When  the  root  cannot  be  extracted,  it  must  be  indi- 
cated as  in  definition  16. 

EXAMPLES. 

1.  Required  the  ^th  root  of  266a*x^, 

4     8 

v/256  =  db4,  the  co-efficient;  a'^x'^^ax^  the  literal  part. 
Hence,  the  root  required  is  4}ax%  or  — 4aa?^. 

2.  What  is  the  5th  root  of—baH^'^l 

The  5th  root  of  5  is  a  surd,  and  must  be  indicated 
1 
thus:   y   5  or  b^. 

Hence,  the  root  required  is  —5'^a^b^  or — a^b^^6. 

*  The  odd  roots  have  the  same  signs  as  their  powers ;  but  the  even 
roots  of  positive  quantities  may  be  either  positive  or  negative.  The 
even  roots  of  negative  quantities  are  impossible.  These  impossible  or 
imaginary  roots  frequently  become  the  subject  of  important  investiga- 
tions, as  will  appear  in  the  sequel  of  this  work. 


EVOLUTION.  37 

3.  Required  the  square  root  of  16a'^b\ 

Eesult,  ±:4^abK 

4.  What  is  the  square  root  of  626oc^y^1 

Ans.  d[=25a?y. 

5.  Required  the  cube  root  of  — 12oa^b", 

Result,  —5a^b. 

6.  What  is  the  5th  root  of—3'2aH'c^^1 

Ans.  — 2a^bc^, 

7.  What  is  the  3d  root  of  7^y?  Ans.  x^^Ky^    ■ 

8.  What  is  the  4th  root  of  Sla'b^l 

Ans.  ±i3as/b%  or  dzSab^b. 

9.  Required  the  5th  root  of —24<3a'b^ 

Result,  — 3aby/b. 

Case  2. 

39.  To  extract  the  square  root  of  a  compound  quantity. 

Arrange  the  terms  according  to  the  dimensions  of  some 
letter,  beginning  with  the  highest. 

Take  the  square  root  of  the  first  term  for  the  first  term 
of  the  root,  and  subtract  its  square  from  the  given  quan- 
tity. Double  the  root  thus  found,  for  a  defective  divisor, 
divide  the  first  term  of  the  above  remainder  by  this  defec- 
tive divisor,  and  annex  the  result  both  to  the  root  and  to 
the  divisor. 

Multiply  the  divisor  thus  completed  by  the  term  of  the 
root  last  obtained,  and  subtract  the  product  from  the  for- 
mer remainder. 

Divide  the  first  term  of  the  remainder  as  before,  for  the 
next  term  of  the  root ;  add  this  last  and  the  preceding 
term  of  the  root  to  the  last  complete  divisor  for  a  new 
divisor.     Multiply,  subtract,  and  proceed  as  before. 

J^ote, — In  these  additions,  regard  must  always  be  paid 
to  the  signs  of  the  quantities. 
4 


38  EVOLUTION. 


4^3  + 1 2a5 + %^—20ac—30bc + 2bc%^a + 36—5c 


4a+36         \12a5  +  9Z>2 
36— 5cll2a6  +  962 


4a+6^>— 5c  \— 20«c— 306c  +  25c2 

/— 20ac— 306c+25c3 

2.  Required  the  square  root  of  x'^'\-2xy+y^. 

Result,  x~{-y. 

3.  Required  the  square  root  of  a*4"4*^^+^^^+'^<^4■l• 

Result,  a2+2a+l. 

4.  Required  the  square  root  of  x* — '^ax^+Qa-x^ — i^a'x 
4- a*.  Result,  x^ — ^ax+a"-, 

5.  What  is  the  square  root  of  16 x' +  24^x^  +  89 x^+ 60 x 
+  100.  Ans.  4a?2+3a?  +  10. 

6.  What  is  the  square  root  of  4a^ — 12a^x+ba^x^+6ax'^ 
+a?*'?  '  Ans.  2a'—3aX'^x'', 

7.  What  is  the  4th  root  of  a'+12a''b+54^a'b^+lOSab^ 
+  816*'?^  Ans.  a+3b. 

8.  What  is  the  4th  root  of  a^^4<a^b^  +  6a'b^-^4^a^b^ 
+6«'?  Ans.  a^—b\ 

Case  3. 
40.  To  extract  any  root  of  a  compound  quantity. 

Arrange  the  terms  as  before  directed  5  take  the  root  of 
the  first  term  for  the  first  term  of  the  root,  and  subtract 
its  power  from  the  given  quantity. 

Take  for  a  divisor  twice  this  root,  three  times  its  square, 
four  times  its  third  power,  five  times  its  fourth  power,  etc., 
according  as  the  2d,  3d,  4th,  5th,  etc.  roots  are  required 
to  be  extracted  5  divide  the  first  term  of  the  remainder,  and 
annex  the  result  to  the  root  before  found. 


*The  4th,  8th,  16th,  etc.  roots,  may  be  obtained  by  2,  3,  4,  etc. 
extractions  of  the  square  root. 


EVOLUTION. 


39 


Involve  the  whole  root,  thus  obtained,  to  the  given 
power,  and  subtract  from  the  given  quantity. 

Divide  the  first  term  of  the  last  remainder  by  the  former 
divisor,  annex  the  result  to  the  root,  and  proceed  as  before. 


1 


« 

1 


^ 

CO 

^ 

CO 

+ 

00 

H- 

e 

o 

53 

CM 

1—1 

C^ 

Jl 

1 

V--' 

A 

'^ 

o 

t<5 

^ 

r* 

« 

*^ 

CO 

^ 

1 

-•a^ 

tH 

1 

O 

^ 

^ 

?3 

1—1 

rH 

00 

+ 

+ 

^ 

S3 

CO 

CO 
05 

C^ 

+ 

4- 

+ 

^^ 

c^ 

e» 

S:> 

^0 

rO 

00 

00 

O 

i-H 
1 

o 

1 

+ 

i 

i 

^ 

53 

e 

CO 

^ 

Tf* 

CO 

T? 

n 

Tf< 

+ 

r-t 

1 

r<5 
CO 

1— i 
1 

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i 

+ 

CJ 
5S 

^i 

o 

Q      - 
00 

1 

2. 

Jl 

00 

C3 
00 

o 

w' 

^ 

<5 

•<5 

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t- 

l> 

®ta 

i> 

^ 

C< 

C<1 

C^ 

C<1 

CO 

+ 

+ 

rH 

+ 

O) 

tM 

<M 

i-£h 

^CJ 

^O 

•+->      , 

e 

e 

rf( 

tJ< 

+ 

•«o 

+ 

+ 

^    1 

<b 

^ 

SP 

^ 

^e.' 

"Is 

53 

CO 

53 

?2 

^1 

CO 
CO 

CO 

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CO 

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CO 

t. 

^^. 

t 

CO 

+ 

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C^^ 

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00  00 

1—1 

00 

00 

40  E\rOLTJTION. 

The  reason  of  this  process,  as  well  as  of  that  laid  down 
under  the  former  case,  is  readily  seen  if  we  consider  that 
in  the  involution  of  powers,  the  second  term  of  the  power 
consists  of  the  continued  product  of  the  index  of  the 
power,  the  first  term  of  the  root  raised  to  the  next  inferior 
power,  and  the  second  term  of  the  root.  Thus,  if  we 
raise  a-{-b  to  the  2d,  3d,  4th,  etc.  power,  we  find  the  second 
term  2ab,  Sa^b^  4a-^Z>,  etc.,  and  the  first  term  of  the  power 
consists  of  the  first  term  of  the  root  involved  to  the  power 
in  question. 

2.  Required  the  3d  root  of  21x^—54^x''y  +  36xy"—Sy\ 

Ans.  3a? — 2y. 

3.  What  is  the  3d  root  of  a?'^— 60?^+ 15a?*— 20a?^+15a?'2 
—6a?  -f- 1'?  Ans.  a?^— 2a?  + 1 . 

4.  Required  the  5th  root  of  32a?5— 80a?*+80a?^— 40a?2-f 
lOo?— 1.  Result,  2a?— 1. 

5.  Required  the  4th  root  of  16a^—96a^x+216a^x''- 
216«a?3  +  81a?*.  Ans.  2a— 3a. 

6.  Find  the  5th  root  of  a?^^+10a?9+35a?3+40a?7— 30a?«— 
68a/5+30a?*+40a?3— 35a?^+  lOo?— 1. 

Result,  a?H2a?— 1. 

41.  J^ote, — The  roots  of  compound  quantities  may 
sometimes  be  obtained,  with  great  facility,  by  a  method  of 
trial.  To  effect  which,  it  may  be  observed,  that  the  power 
of  a  binomial  consists  of  as  many  terms  as  there  are  units 
in  the  exponent,  increased  by  one.  Hence,  if  the  number 
of  terms  of  a  quantity,  whose  root  is  to  be  extracted,  ex- 
ceeds this  sum,  we  may  conclude  that  the  root  consists  of 
more  terms  than  two. 

When  the  root  is  judged  to  be  a  binomial,  take  the  roots 
of  the  extreme  terms,  and  connect  the  results  by  the  sign 
+  or  — 5  but  when  the  root  appears  to  contain  more  terms 
than  two,  take  also  the  roots  of  one  or  more  of  the  other 
given  terms,  and  connect  the  roots  thus  obtained  by  the 
signs  +  or  — ,  as  may  be  judged  proper. 

Involve  this  supposed  root  to  the  given  power,  and  if 


ALGEBRAIC  FRACTIONS.  4fl 

the  result  agrees  with  the  quantity  given,  the  root  is  mani- 
festly correct. 

This  expedient,  however,  is  not  likely  to  abridge  the 
labor  of  a  student  in  the  early  periods  of  his  course. 


Section  III. 

ALGEBRAIC  FRACTIONS. 

42.  Algebraic  fractions  depend  upon  the  same  princi- 
ples as  those  in  common  arithmetic,  and  are  managed  by 
similar  rules,  proper  regard  being  paid  to  the  signs  and 
algebraic  modes  of  operation. 

EXAMPLES. 

1.  Required  the  greatest  common  divisor  of  cc^ — b^Xy 
and  x'^+^bx+bK* 

x^—b''x)x^+2bx+  b\\ 


25a: +263 
x-\-b)x'^ — h\x — h 
x^-{-bx 


—bx--b^ 
—bx—b] 
Hence,  x+b  is  the  divisor  required. 


*  In  examples  of  this  kind,  we  are  frequently  obliged  to  adopt 
expedients  which  are  not  usual  in  common  arithmetic.  When  every 
term  of  the  divisor  has  a  factor,  which  does  not  contain  a  letter  or 
number  common  to  every  term  of  the  dividend,  we  divide  by  it,  and 
use  the  quotient  as  a  divisor.  When  the  divisor  has  been  thus  re- 
duced, it  often  happens,  that  the  division  is  still  impracticable ;  in 
4* 


42  ALGEBRAIC  FRACTIONS. 

3^2 2a 1 

2.  Required  to   reduce      ^ ^ ^g  '  ,  -i  ^^    its    lowest 

terms."^ 


First  to  find  a  common  divisor. 


3a2_2a— -1)  4a^— 2a3—  3a+  1 
3 


12a^— 6^2—  9a.+   .3(4a 


2a3_  5a  +  3 
3 


6a^—15a+  9(2 
6a3_  4a_  2 


— lla+11 


which  case,  we  multiply  the  dividend  by  such  a  number  as  will  make 
its  first  term  a  multiple  of  the  first  of  the  divisor. 

By  these  processes  we  make  no  change  in  the  common  divisor,  un- 
less we  divide  by  a  number  or  quantity  which  is  common  to  the  given 
terms ;  or  introduce  into  one  of  them  a  divisor  of  the  other. 

*  Solutions  may  sometimes  be  facilitated,  by  resolving  the  quanti- 
ties concerned  into  their  component  factors.  In  this  example  the 
numeratoris  evidently =3^^ — 3a-^a — l=3a  (a — 1) -f  a — l=(3a-|- 1) 
(a — 1);  and  the  deuomin?i.tor=^Aa^—4a^-\-2a^—2a^a'\-l=4a^{a—l) 
Hr2«(a— l)--l(o— l)=(4a24-2a— l)(a— 1).  Hence  a— 1  is  mani- 
festly the  common  divisor,  and  '    ^  .  ^ :  the  required  result. 

In  example  3,  we  have  the  numerator=  o3(a2 — l!^),  and  from  ex- 
ample 8,  page  25,  it  is  seen  that  a'^—'b^=(^a^—b^){a^J^h2),    Hence, 

ai — J2  IS  the  common  divisor ;  and  — ;; — — •=  „  ,    -- 


ALGEBRAIC  FRACTIONS.  43 

Com.  div.  a —  l/Sa^ — 3a 


3.  Reduce— — rr— to  its  lowest  terms.     Result, r-     '^■- 


4.  Reduce ; —  to  its  lowest  terms.  Result, 

xy+y  y 


P,a^+10a*b  +5a^b^  .     , 

5.  Reduce  --j—, — o  ^i:^  ■  n  ?,■.  .  j.^  to  its  lowest  terms. 
a^b-^  2a^b^-{'2ab^+b'^ 

^      ,       6a^+da^b 
Result,  ■ 


^a^b^ab^+b^ 


6.  Reduce  — -to  its  lowest  terms. 

a^ — a^x — ax^ + x^ 

Result, 


^  ^   ,      6a'' +  lax—-3x^      .     , 

7.  Reduce^— ——r rrrrto  its  lowest  terms. 

6a^+llax-\-3x^ 

Result,  TT — ; — 
^  3a+x 

8.  Reduce   ^-    ,  ^  „  .  ^ to  its  lowest  terms. 

Result,  ^ 


'3a?2+2a?+l 


44?  ALGEBRAIC  FEACTIONS. 

(I     2c     3x 
9.  Reduce  oT?    q3  7~>    to    other    equivalent    fractions 

having  a  common  denominator.* 

6a^d      Sabc.      %dx 
'12abd'  nabd'  T2M 


10.  Reduce  t-^-tt'-,  and  ^r-  to  a  common  denominator. 

9b       Sac      6ad 
^^^   '12^2'   12a^'  12^ 

1    3a^  2a^4-5'^ 

11.  Reduce  TT,— 7- ,  and  7-;  to  a  common  denomi- 

nator. 

4a  4-  45    9a^  +  9a25    24^a''+12b^ 


Result, 


12a+126'   12a  +  126'   12a +126 


*  The  least  common  denominator  should  always  be  foand  in  exam- 
ples of  this  kind. 

To  find  the  least  common  multiple  of  two  or  more  numbers  or  quanti- 
ties, divide  such  of  them  as  have  a  common  divisor,  by  the  greatest 
prime  number  they  contain.  Repeat  the  process  on  the  quotients  and 
undivided  quantities  as  long  as  a  divisor  common  to  two  of  them  can  be 
found ;  then  multiply  these  divisors,  the  several  final  quotients,  and 
undivided  quantities  continually  together,  the  product  is  the  common 
multiple  required.  For  such  product  must  contain  all  the  factors  in- 
cluded in  the  given  quantities,  and  none  of  them  are  repeated  oftener 
than  in  the  given  terms.  Thus  to  find  the  least  common  multiple  of 
4a^,  Sab,  66^,  IQbc.  The  only  common  measures  of  the  co-efficients 
are  2  and  3,  and  the  literal  divisors,  common  to  two  or  more,  are  a 
and  b.     Arranging  them  thus  : 

2)4a3,  3ab,  652,  105c 

2a3  Sab  Sb'^  5bc 
2^3  ab  b^  5bc 
2^3  b  53  5bc 
2«3X   1  XiX  5c  X  2X3c5=60a3i%  the  quantity 


sought. 


ALGEBRAIC    FRACTIONS.  45 

12.  Reduce  ttt^-tt  j  -,  to  a  common  denommator. 

40    3  a     ' 

93a    80aa?    120a^  +  240a? 
•^^^"   '  120^'   120^         120a 

2a7 

13.  Reduce  a — Trrto  an  improper  fraction. 

3^—207 

Result,  — 

3a 

2a7 8 

14.  Reduce  10 -| — ^ —  to  an  improper  fraction. 

32x— S 
Result,  — ^ • 

15.  Required    an    improper    fraction    equivalent    to 
l+2a:—  ^  .  Result, ^^ 

16.  What  is  the  equivalent  improper  fraction  to  a — 

.22 — a^  2a- — z^ 

7  Ans. 

a     '  a 

3a3? '  1  4^?^ 

17.  Reduce — —  to  a  whole  or  a  mixed  quantity. 

Result,  3a? -4 

a-^x 

2a3+252 

18.  Reduce  -. —  to  a  whole  or  mixed  quantity. 

4Z'2 

Result,  2a +25+ =. 

a — b 

^^    ^    ,        27a3— 36^—40? +9a2  ^ 

19.  Reduce ^— to  a  mixed  quantity. 

Result,  3«+l-?^^ 


46  ALGEBRAIC  FRACTIONS, 

20.  Eeduce — — to  a  whole  or  mixed  quan- 

tity. 

_,      ,                    '^ax^' — h 
Result,  x^ — a^  -\ 

21.  Add  ^  3^r  and  ^.^  together.  Sum,  -^^^— 

22.  Required  the  sum  of  a — -r-  and  h-\ ■. 

_       ,           _     2ahx — 3c^2 
Result,  a-\-b-\ ^ • 

^  _i_  3  2a 5 

23.  What  is  the  sum  of  2a+ — ^,  and  ^a-\ — ^ 

14«— 13 
Ans.GaH ^^j — 

24.  Add  — 1-,  to — —,.  bum  — - — 7—  or  2+-; — ji 

a — b       a-\-b  a^—b^  a\ — 6-, 

a  a — X 

25.  What  is  the  sum  of  2a, and  — --1 

^  a — X  a-\-x 

Sx^ — ax 
Ans.  2a+2+  ^^3^ 

^       Sa  4)a  -n.        .    1      ^^ 

26.  From-x-  subtract-y-  Remainder,f)g 

^       Sax     ,     2ax  ^         ax 

27.  From-^-  take  -tt  .  Rem.  j^ 

2a — b  Sa — 46 

28.  From  — -. —  subtract  — -^t — . 

Qab—3¥—12ac+16bc 
Kem. j23^ 


ALGEBRAIC  FRACTIONS.  47 

29.  From  4a -{ — - —  subtract  2a — ■^-— — ^ 

lOa^ — 5a6 — 7ac  + 1  lie 
Eem.  2«+ ^^^^^^ 

a-\'CO  a — X  4a? 

30.  Subtract^-; .  from  a-{-  — — — r     Kem.  a- 


a(a — x)  a(a  +  xy  a^ — x^ 

31.  Required  the   continued   product   of  -r  -q-    and 
a — X  _,     -     ,   2a — 2a? 

32.  Required  the  product  of    .       ^  and  — g^v. 

Prod.      "^ 


'  «6+c2 

33.  Required    the    continued    product   of  a r  "TTT 

3a?         ,       76  ^     ,  216 

75  and  7- r,  Frod.-y — 

a — 0  ox — a?^.  b — x 

34.  Required  the  continued  product  of— 7-7-7    — r^' 

fi> -J"  c?       ax  ~Y~  X 

ax 

and  0?+ Prod.  2a^ — ax — 2a64-6a? 

a— —a?* 

3D.  Divide  ^  by  ^^-.  Quotient,g- 

36.  Divide   o"- — by-7r^ — .  Quot. —^  - 

Soay    ^   2oyz  la 

37.  Divide  a by  a+- .  Quoi. - 

a  +  x     ^  a — X  a^  -i~ax 

38.  Divide   -— -  by  -— -. — --7,  Quot.  1 


48  ALGEBRAIC  FRACTIONS. 

on     n-   'A     3^— li.     1-^+^  n      r^^ 

39.  Divide  ^^-by-^^.  Quot.  -^^^ 

40.  Divide  -vrT^  by-Tj-r  Quot.  —pr — . 

12cd     ^  bd.  ^          I0ac2 

^1-  ^^^^^^  2a^-4aH~2^^  ^^  "4-^-4T-* 


Quot.  2a +  ' 


a 


42.  Divide  ^z:^^-j^-^^y -z:^.  Quot. a:+~ 

Quot.  a3+2a2a?+2aa?2+.r'^ 

13  4  4 

44.  Divide  a:* — -^x'^'\-x^'\--kx — 2byg^ — 2. 

3         1 

Quot.  jo?"^ — 2^^+l. 


*  By  inverting  the  divisor,  and  resolving  the  terms  into  their  com- 
ponent factors  we  have — 

2{a—b)  (a—b)  ^  5a(^b) 

"Whence  by  expunging  the  corresponding  factors,  from  the  numerator 

and  denominator  we  find  the  fraction  reduced  to —  2a  + 

a  '      a 

In  like  manner — 

a:^— 3^  a:~h   __{x^  +  b^Xx  +  h)(x—b).(a;—b)_x2  +  b^ 

x^—2bx  +  b^Xc^-\-bx  '(x—b){x^b),x(^x'fb)       ~       ^ 

=  X-^ 

'      X 


ALGEBUAIC  FK ACTIONS.  49 

45.  Eequired  the  4th  power  of  -3^.  Result,  ^^ 

a — b 

46.  Required  the  3d  power  of  -^r^ 

Result  1- 


'^a^  +  3a^b-{'3at^-\-b^ 


2x 
47.  What  is  the  4th  power  of  1 — —7 


a 


Sx    24a?2     32a:3     I6x^ 
Ans.  1--+-^--^+-^ 

64a?^  4  J? 

48.  What  is  the  cuhe  root  of  ^7^'^  '^^^'"Sa^ 

49.  Required  the  square  root  of  J^il^- 

3a^ 
Result, 


2xy^ 


11         3 

50.  Required  the   square  root  of  x^—Sx^^-^x^—^x 

1  3       1 

+_.  Result,  a^'— 2^4-j 

1         1         41         43 

51.  What  is  the   3d  root   of  go^s— -a:^ +— a:*— — a:^ 

41         1        1  .1^1 

52.  Required  the  square  root  of  a^+2b. 

b      b^      b"       55* 
Result,  a+~^3+^^g^,  etc. 

5 


50  EQUATIONS. 

Section  IV. 
EQUATIONS. 

43.  An  equation  is  an  algebraic  expression,  indicating 
that  one  quantity,  or  combination  of  quantities,  is  equal 
to  another.  The  quantity,  or  combination,  which  stands 
on  either  side  of  the  sign  of  equality,  is  called  a  member 
of  the  equation. 

Thus,  ax-]-bx=ah-\-cd,  is  an  equation  of  which  ax-\-bx 
is  the  first  member,  and  ah-\-cd  the  second. 

44.  In  the  application  of  algebra  to  the  solution  of 
problems,  the  quantities  concerned,  both  those  which  are 
given,  and  those  which  are  required,  are  usually  repre- 
sented by  symbols,  and  the  conditions  of  the  problem 
translated  into  the  language  of  algebra.  One  or  more 
equations  are  thus  formed,  including  unknown  as  well  as 
known  quantities. 

45.  The  number  of  independent  equations  required  for 
the  solution  of  a  problem,  is  equal  to  the  number  of  un- 
known quantities  employed. 

46.  A  simple  equation  is  one  which  includes  only  the 
iirst  or  single  power  of  the  unknown  quantity  or  quanti- 
ties, as  ax+h=^cx — d, 

47.  A  quadratic  equation  is  one  whiclr  includes  the 
square  of  the  unknown  quantity.  An  equation  which 
contains  no  power  of  the  unknown  quantity,  but  the 
square,  is  called  a  simple  qnadraticy  as  ax^=^h;  but  when 
the  square  and  simple  power  of  the  unknown  quantity  are 
both  included,  it  is  called  an  adfected  quadratic  equation; 
as  x^ — ax=bc, 

48.  A  cubic  equation  is  one  which  contains  the  cube  of 
the  unknown  quantity;  as  o(f^-{-aX'-i-bx=c, 

49.  A  biquadratic  equation  is  one  which  contains  the 
fourth  power  of  the  unknown  quantity;  ns  x^-\'aV-{-c'''X 
=^bH\ 

50.  Equations,  in  general,  are  said  to  be  of  as  many 


SIMPLE    EQUATIONS.  51 

dimensions  as  there  are  units  in  the  index  of  the  highest 
power  of  the  unknown  quantity. 

51.  In  general,  an  equation  which  contains  but  one 
power  of  the  unknown  quantity  is  called  a  pwe  equation; 
thus,  x'^—-b-\-c^  is  called  a  pure  quadratic  equation;  x^=b, 
a  pure  cubic  equation,  etc. 

Great  part  of  the  business  of  algebra  consists  in  de- 
ducing, from  given  equations,  the  values  of  the  unknown 
quantities  which  they  contain.  This  is  effected  by  making 
correspondent  changes  in  both  the  members  of  the  equa- 
tion, so  as  to  obtain,  eventually,  the  unknown  quantity  on 
one  side  of  the  sign  of  equality,  and  known  quantities  on 
the  other. 


SIMPLE  EQUATIONS. 

Containing  one  unknown  quantity. 

To  facilitate  the  solution  of  equations  generally,  the  fol- 
lowing principles  or  modes  of  operation  must  be  observed. 

♦  Mode  I. 

52.  Any  quantity  may  be  transposed,  that  is,  removed 
from  one  member  of  the  equation  to  the  other,  by  chang- 
ing its  sign.* 

EXAMPLES. 

Given  '7x-\-5a=:6x-{-3b  to  find  x. 
By  transposing  ba  and  6x,  we  have  Ix — 6x=3b — 5a, 
or  x=3b — Sa, 

*  Hence,  if  any  term  be  contained  in  both  members  of  the  equa- 
tion with  the  same  sign,  it  may  be  taken  from  both;  and  the  signs  of 
all  the  terms  msy  be  changed,  without  destroying  the  equation. 


52  SIMPLE  EQUATIONS. 

2.  Given  Sy— 3=:7i/+5  to  find  y.  Result,  2/=8. 

3.  Given  15a?— 6  =140? +  20,  to  find  x.     Result,  a?=:26. 

4.  Given  20— 132^=45— 14y+ 4,  to  find  y. 

Result,  ?/=29. 

5.  Given  8a?+  15=15+907—17,  to  find  cc. 

Re  :ilt,  a:=17. 

6.  Given  25a? — 5=24a?  +  26 — c,  to  find  a?. 

Result,  a?=  36 — c. 

7.  Given  36— 9a?=76— 8a?— 4J,  to  find  a?. 

Result,  a?=4c^ — 46. 

Mode  2. 

53.  The  CO- efficient  of  an  unknown  quantity  may  be 
taken  away,  by  dividing  every  term  of  the  equation  by  it. 

EXAMPLES. 

Given  ax-{-ba=ad,  to  find  a?. 
By  dividing  by  a,  we  have  x-{-b=d;  hence  by  transpo- 
sition, x=d — b. 

2.  Given  5a?==  25,  to  find  a?.  Result,  a?=5. 

3.  Given  30a?=180,  to  find  a?.  Result,  x=6, 

4.  Given  6a?+ 17=44 — 3a>,  to  find  a?.  Result,  a?=3. 

5.  Given  3aa? — 4a6=2aa? — 6ac,  to  find  a?. 

Result,  a?=46 — 6c. 

6.  Given  3a?- — 10a?=8a?+a?2,  to  find  a?. 

Result,  a?=9. 

7.  Given  6y^—Sy''=22y^—Sy^,  to  find  y. 

Result,  y=3i 

Mode  3. 

54.  An  equation  may  be  cleared  of  fractions  by  multi- 
plying all  its  terms  by  any  common  multiple  of  the  deno- 
minators. 


SIMPLE  EQUATIONS.  53 


_,.  X      X      X  6  ^1 

Given  - — r  +  -=l7T5to  find  x. 
2     3     5        o 

Multiply  by  30,  and  then  we  have  15a? — 10x-\-6x=b5; 
whence  lla?=55,  and  a?=5. 

XXX 

2.  Given  —+—-==:  — +  7,  to  find  x.  Result,  x=12. 

^    ^.        ^—5     ^       284—07        ^    , 

3.  Given  —r — [-Qx= — ^ ,  to  find  x. 

Eesult,  x=9, 

^.  11 — X     19 — X        ^    _  T-,      , 

4.  Given  a? -1 ^ — = — ^ — ,  to  find  a?.     Kesult,  a7=::5. 

.    ^.        o       2a?+6     ,     llo;— 37        ,    , 

5.  Given  3a? H — =5-^ ,  to  find  x. 

0  Ai 

Result,  a?=7. 

Mode  4. 

55.  When  the  unknown  quantity  is  under  a  radical 
sign,  such  sign  may  be  taken  away,  by  first  transposing 
the  terms,  so  as  to  leave  the  radical  quantity  alone  on  one 
side  of  the  equation,  and  then  involving  each  member  of 
the  equation  to  the  proper  power. 

EXAMPLES. 

Given   ^^+40=10— ^/a?,  to  find  a?. 
Squaring  both  sides,  a?+40=100 — 20  v/a?+a?. 
By  transposition,  20^/^=60,  or  ^/a?=3. 
Squaring  both  members,  a:=9. 

2.  Given  x/a?  +  l— 2:rr3,  to  find  x.  Result,  a?=24. 

3.  Given  4^3^+4  +  3=6,  to  find  x.  Result,  a?=7f . 

4.  Given  sf  x — 16  =  8 — sf  x^  to  find  x.      Result,  a?=25. 

5* 


54?  SIMPLE  EQUATIONS. 


5.  Given  v/4a4-a7=2V^+^ — V^j  to  find  a?. 

Eesult,  0?= — ^ i — 

2a — b 


6.  Given  's/4>a^+x^=:i^U^+x*,  to  find  x. 

Result,  a:=V-2^ 


7.  Givena?  +  a+\/2aa?-[-^^=^  to  find  x, 

b^—2ab+a^ 

Result,  X= pry 

Mode  5. 

56.  When  the  member  of  an  equation,  which  contains 
the  unknown  quantity,  is  a  complete  square,  cube,  etc., 
the  equation  may  be  reduced  by  extracting  the  square  root, 
cube  root,  etc.  of  both  its  members.* 

EXAMPLES. 

Giv^n  x^—6x^+  12a?— 8=  343,  to  find  x. 
Extracting  the  cube  root,  x — 2=7,  or  x=9, 

2.  Given  a?24.4a?4-4=81,  to  find  x.  Result,  a?=7» 

1     81 

3.  Given  ^H^+T=-r'  ^^  ^^^  ^*  Result,  a?=:4?. 

4.  Given  x^+Sax'^+Sa^x+a^^b^^  to  find  x. 

Result,  x=b — a 

5.  Given  9a?2+17=:98,  to  find  x.  Result,  x=3. 

6.  Given  a?*— 4a?^-f  6a?2~-4a?+l  =  256,  to  find  x. 

Result,  a?  =5 

7.  Given  3a?5=96,  to  find  x.  Result,  x=% 

*  These  modes  of  reducing  equations  depend  upon  a  few  self-evi- 
dent truths.  When  two  quantities  are  equal,  and  equal  quantities  are 
added  to  or  subtracted  from  both,  or  both  are  divided  cr  multiplied  by 
the  same  number,  or  the  same  power  or  the  same  root  of  each  is  taken, 
the  results  in  each  case  are  equal  to  each  other. 


SIMPLE  EQUATIONS.  55 


PEOMISCUOUS  EXAMPLES. 

1.  Given -0? — 2;a?+-a7=-,  to  find  x.  Result,  x=l^ 

4a 


2.  Given  s/x-^-  ^2a+x= —         ,  to  find  a?. 

Result,  x=-^a 
o 

Result  x=-xy/ — r— 
Z        0 


4".  Given  a  +  x=  ^/ a^+x s/ ^^^-i- x\  to  find  x. 

Result,  x-= a 


5.  Given  2+ v'3a?=\/4<+5a7,  to  find  a?. 

Result,  x—\% 

6.  Given  a — a?= ,  to  find  a?.  Result,  x=~ 

a — X  '        2 


7.  Given  V«+3?+V«^ — ^=  \/aa:,  to  find  a:. 

4a^ 
Result,  0?=-- — 7 

a  a  5 — 2a 

8.  Given  --; — l-:; =5,  to  find  x.  Result,  a:=  V  — t — 

^    ^.       7a?+16       a?+8      a?        ,    , 

9.  Given -jj ^___^^,  to  find  a.. 


^_     ^.        4a?  +  3  ,  7a?— 29     8a?+19  ^    .    , 
10.    Given^^+g^-^=-^g-,tofinda?. 


Result,  a?=8. 

ind  X, 
Result,  a?=6. 


56  SIMPLE  EQUATIONS. 

11.    Given  — ^r—  : — - — :  :7 : 2,  to  find  a?.* 

<w  0 


Eesult,  a:=3/g 


ijto  find  0?. 


12.  Given   \/5+a7+\/a?=:\/5+a? 

Result,  a?=4'. 

13.  Given  a?*4-2a:2+ 1=4225,  to  find  x.      Result,  a?=8. 

14.  Given  a?s_9a?*+ 270-2— 27=2197,  to  find  x. 

Result,  a?=4. 

15.  Given  a?^— 17=130— 20?^,  to  find  x.     Result,  a?=7. 

16.  Given  a?^— 100=412— 7a:3,  to  find  x.  Result,  a?=4. 

17.  Given  a?*— 4a?^+6a?2—4a?+ 1=4096,  to  find  x. 

Result,  a?=9. 

18.  Given  x^—^ax''-\'l^a^x—'^a?—h%  to  find  x. 

Result,  a?=62_|.2«. 

19.  Given   625a:4— 1000a?3+ 600072— 160a? +16  =  256a?S 
to  find  X,  Result,  a?=2. 

20.  Given    ^la^x''+b^ha^x'^-\-^^b^ax-\'^¥=12bc^xS    to 
find  X. 

Result,  X—-Z — 

'        5c — 3a 


21.  Given  x-\-  s/a — x=^\/  a- — x  to  find  x. 

Result,  x=a — 1. 


22.  Given    V4^+ Va?*— ^^=a?— 2,  to  find  a?. 

Result,  x=2\ 

*  If  four  numbers  are  proportional,  the  product  of  the  extremes  is 
equal  to  that  of  the  means ;  hence  an  analogy  is  readily  converted  into 
an  equation.     Vide  Article  62. 


SIMPLE  EQUATIONS.  57 

4      . 


23.  Given    V24-a?+v^a?=N/2+a?  to  find  x. 

Result,  a?=S 


Section  V. 
SIMPLE  EQUATIONS. 

Containing  two  unknown  quantities. 
Rule  I. 

57.  Multiply  or  divide  one  or  both  equations,  by  such 
numbers  or  quantities  as  will  make  the  co-efficients  of  one 
of  the  unknown  quantities  the  same  in  both  equations. 
Then  take  the  sum  or  difference  of  these  equations,  accord- 
ing as  the  signs  of  the  corresponding  terms  are  different,  or 
the  same ;  the  result  will  be  a  new  equation,  containing 
but  one  unknown  quantity.*  The  value  of  the  remaining 
unknown  quantity  may  then  be  determined  by  the  methods 
in  the  last  section. 

EXAMPLES. 

Given  4r+2/=34«,  and  ^y-{-x=lQ^  to  find  the  values  of 
X  and  y» 

First,  to  eliminate  a?,  we  multiply  the  second  equation 
by  4^ ;  whence,  16y+4a?=64'.  From  this  we  take  the  first 
equation  and  15y=30;  whence  y=2. 

The  value  of  y  being  now  obtained,  we  find  x— 16 — 4?/ 
=rl6— 8=8.  ' 

Or  to  eliminate  a?,  divide  the  first  equation  by  4,  and 
1        34 
we  shall  have  ^+"T'3^~X'  which   subtracted  from   the 

second  equation,  leaves —y=—5  whence  y= 2,  as  before. 

*When,  by  this  or  any  other  process,  a  quantity  or  letter  is  caused 
to  vanish  from  the  equations,  it  is  said  to  be  eliminated. 


58  SIMPLE    EQUATIONS. 

2.  Given  ax-\-byt=m,  ex — dyz=n^  to  determine  the 
values  of  x  and  y» 

To  eliminate  a?,  multiply  the  first  equation  by  c,  and  the 
second  by  a,  whence  acx-\-bcy=^cm^  and  acx — ady=^an; 
by  subtraction,  bcy-^ady=cm — an;  and  by  dividing  by 
cm — an 

Or,  to  eliminate  y,  multiplying  the  equations  by  d  and  b 

respectively,  adx-{-bdy=dm,   bcx — bdy=bn;  adding  these 

"  dm  +  bn 

equations,  adx-\-bcx=^dm-\-bn;  whence  a?=     .     , 


* 
oc' 


6.    uiven  ^  g^^3y^39^  to  find  the  values  of  a?  and  y. 

Result,  a:=3,  y=5. 

,     p.        (  15a?+34y=241, 

^-    ^iven  ^  18a?— 172/=  58,  to  find  the  value  of  x  and  y. 

Result,  07=7,  2/=4. 
p.     p.        C  6a7 — 4^y=34, 

0.    uiven  ^  9^_2a?=27.         Required  the  values  of  x 
and  y.  Ans.  a?=9,  y=5. 

^'^^^  ^    4^ — lyz=^^  to  find  the  values  of  a?  and  y. 

Result,  a?=8, 3/=4<. 


7.   Given . 


X       y 
2^3 


X       y 
-o'  +  "o"=8)  to  find  X  and  y. 


Result,  a:=6,  y=12. 


*The    known  quantities  being  expressed  in  general  terms,  this 
solution  is  applicable  to  the  following  exanaples. 
Thus  if  in  the  6th  example  we  make — 

700  +  12 
af=ll,  ^=3,  m=^100,  c=4,  d=l,  w=4,  «=rz— ^77^=8, 

77   -7- 1-* 
___400— 44__ 

^"~77  4-12""^* 


SIMPLE  EQUATIONS. 


59 


S.    Given. 


15     20 


y_ 
1 10 


30 


=  1,  to  determine  x  and  y. 


Result,  3?=:  30,  ?/=20. 


Rule  2. 

58.  Find,  from  each  equation,  the  value  of  one  unknown 
quantity  in  terms  of  the  other  quantities* 

Assume  the  values  thus  obtained,  as  the  different  mem- 
bers of  a  new  equation.  The  value  of  the  unknown  quan- 
tity contained  in  this  equation  may  then  be  determined  as 
in  the  last  section. 


EXAMPLES. 


Given 


707+^=51, 


9x — -|-  =  61i,  to  find  the  values  of  a?  and  y. 

o 


From  the  first  equation  y=  357 — 49a7. 
From  the  second  equation  y='72x — 490, 
Hence,  72a:— 490=357— 49a7, 
72a:+49T=357+490, 

_847_ 

^-121-^' 

y=  357— 49a:=  357—343  =  14. 


2.    Given 


and  y. 


y  +  72/=99 


7a7+-^=51. 


Required  the  values  of  x 
Ans.  a?-^7,  y=14. 


60 


SIMPLE  EQUATIONS. 


3.    Given  , 
X  and  y. 


^+2     ^ 


2/  +  5 


+  10a?=  192.       Eequired  the  values  of 
Ans.  07=19,  ^=3. 


4.    Given 


5.    Given , 


X       y 
-=r — oT~'^*     Quere  the  values  of  x  and  y  1 

Ans.  a?=20, 3/=:18. 

3  "^  5  ~^^'      . 

3a?     2y     67     ^  .        , 

-^+-o-="3'*    Quere  the  values  of  x  and  y  ? 


Ans.  07=15,  y=20. 

an-         (  ^ — ^y=^i 

^^^^  ^  x:y::a:b.     Quere  the  values  of  o?  and  y? 

bd 


ad 
Ans.  0?= ;tt,  y 


a_^26'  ^"~«— 26 


7     Given 
and  y? 


(x—2y     X  _3x-^y 
4     "^  5 "~     5     ~^^' 


0?  -4- 1/        T       1] 

— ~A — =r^=8|.    Quere   the   values  of  x 
4     '     7  'J     ^ 


Ans.  07=20, 2/= 8. 


Rule  3. 


59.  Find  in  either  equation,  the  value  of  one  unknown 
quantity,  and  substitute  the  value  thus  found  in  the  other 
equation  j  whence  will  arise  a  new  equation,  containing 
but  one  unknown  quantity,  whose  value  may  be  determined 
as  already  taught. 


SIMPLE    EQUATIONS.  61 


EXAMPLES. 


C  V  ^ 

\    0?+— =12  f 
Given  <  2  >  to  determine  x  and  y. 

(  7a?— 3y=:58  ) 

From  the  first  equation  x— 12^—-^.     This  value  being 
substituted  for  x  in  the  second  equation,  we  have 


xMultiplying  by  —2,  Ty  +  G^/— 168=— 116. 
13 


Whence  ?/= =4, 


a?=:12— 1-=12— 2=10. 


2     Given  ^    ^^—  %=15^' 

z.    uiven^  10a?+152/=825.     Quere   the   values   of  a? 

and  yl  Ans.  a?=45,  y=25. 


3.  Given, 


3       5"^' 


— -4.^=:31j  to  find  07  and  y, 

Eesult,  a?=  30, 3^=25 

^-  ^^^^^  ^  0? :  2/  ::  5  :  3,  to  find  07  and  3/. 

Result,  07=15,  y=9 


5.  Given 


-^  +  3o?=29,  to  find  x  andy. 


Result,  07=9,  y=:6. 


62  SIMPLE  EQUATIONS. 


Rule  4. 


60.  Multiply  one  of  the  equations  by  an  indefinite  quan- 
tity, and  to  the  equation  thus  formed,  add  the  other  given 
equation. 

Assume  the  sum  of  the  co-efficients  of  one  required 
quantity  equal  to  0,  and  thence  determine  the  value  of 
the  assumed  multiplier.  The  new  equation  will  then 
contain  but  one  unknown  quantity,  which  may  be  found 
as  before. 

iSXAMPLES. 

Given  |  o  Xfi^—  1 S2  (  ^^  ^^^^  ^^^  values  of  x  and  y. 

Multiply  the  second  equation  by  the  indefinite  quantity 
m,  and  we  shall  have  3mx-\-8my=132m, 

Adding  this  equation  to  the  first, 

Smx+Hx+Smy — 9y=132m+69. 

7 
Assume  37?^+7=0,  whence  m= — —^ 

_132^+59_— 308  +  59_924<— 177_ 
^^^y-    Sm—9    "     56     ^     -   56  +  27    ~^- 


-3-^ 

132— 8y     132—72     60     ^^ 
Hence  x= ^—^=  — ^ =—=20. 

9 

Or  assuming  Sm — 9=0,  7n=—^ 

1188     ^„ 
132;7^+59_  8     ^         1188+472_^^ 
^'^^^-      3^2+7  "27  -"     27+56  "~^^- 

8  "^ 


SIMPLE  EQUATIONS. 


63 


^    p.        (  15a?— 171/=  12, 

Z.  triven  I  i7^_i0y.:z.97,  to  find  x  and  3/. 

Result,  a?=ll,  2/=9. 

f  3a74-72/=79, 
3.    Given  j  o^_  -i.T=:9,  to  find  x  and  3/. 

Result,  a?=10,  2/=7. 

,     p.        (    807 — 5i/=60, 

*•    vjiven  ^  102/— 3a?=75,  to  determine  x  and  3/. 

Result,  X— 15,  2/=  12. 


5.    Given 


^      V      ^ 

2         3  6 

_-a:4-_y=l0— ,  to  find  a?  and  y. 


Result,  a?=lO,  ^=16. 

Examples  to  exercise  the  foregoiivg  rules, ^ 

p.         (     X — ?/=c/, 

^  xy=^p,  to  find  a:  and  y. 
Squaring  the  first  equation,  x'^ — 2xy+y'^=^d^. 
Multiplying  the  second  by  4,  4a?y=4p. 
Taking  the  sum,  x^-{-2xy-\-y^=d^-{-4^p. 
Extracting  the  root,  x+y=  ^/ d'^+4}p,(^.) 


d-h\/d^-\-^P 


Adding  the  first,  2x=d-{-  y/d^  +  4<pj  or  x= ^ 

Subtract  the  first  equation  from  equation  j^j  and 

V>  +  4^ — d 


2?/=  V  c?^  +  ^P — d,  or  y = 


*In  the  solution  of  these  and  other  similar  problems,  the  ingenious 
student  will  find  expedients  which  are  not  clearly  indicated  in  any  of 
the  preceding  rules,  by  which  his  labor  may  be  frequently  abridged. 


64f  SIMPLE  EQUATIONS. 

2.  Given  ^  ^+2/=^» 

I  xy=py  to  find  x  and  y. 

Subtracting  4  times  the  second  from  the  square  of  the 

first,  we  have  o?^ — 2xy'{-y'^=s^ — 4j9. 

Whence  x — y = zh  >/  s^ — 4/?/ *  and    _ 5±  s/  5^ — 4p; 


y=szf.^s^ — 4p. 


^  0?^ — y^=n,  to  find  a?  and  t/. 

Dividing  the  second  by  the  first,  a? +  2/=—, 

Hence,  taking  the  half  suiji  and  half  difference  as  before 
n-i-m^  n — m^ 

x=—^ and  2/=-o — • 

^.    Lriven  I    ^3_  ^3^^^  ^^  ^^^  ^  ^^^  ^^ 

Subtracting  three  times  the  first  equation  from  the  se- 
cond, x^ — 3x^y  +  3xy^ — y^=b — 3a, 

Hence,  extracting  the  root,  x — y=4^b — 3a,  which 
put  =c,  (c/^.) 

Divide  the  first  equation  by  this,  whence  xy= — (jB.) 

From  equations  ^  and  J5,  by  proceeding  as  in  the  first 
example,  we  find 

,  (  4a        >        ,  (  4a        > 

oc=h  <  ^c^^ +c  hy=i  I  VC-] c  I 

(  c    '      )  (  c         ) 

Or  divide  the  sum  of  the  given  equations,  by  equation 

a+b 
^  and  a?3  4-2a?y+2/2=:— — , 


*  As  the  square  of  a  quantity  whether  positive  or  negative  is  al- 
ways positive,  the  square  root  is  often  ambiguous;  and  therefore  sus- 
ceptible of  the  sign  -f  or  — .  In  this  example,  the  ambiguity  arises 
from  the  uncertainty  which  re  or  y  is  the  greater. 


SIMPLE  iiQUATlOiXS.  65 

Whence  by  extracting  the  square  root,  57+1/=^/ , 

Then  taking  the  half  sum  and  half  difference  of  this 
equation  and  equation  A^ 

5.    Given)    oT^~^?'a    i:   j  j 

I  x^-\-y^=Oj  to  find  x  and  y. 

From  the  first  a?^+3a?22/+3a:?/2+2/3=:a3. 

By  subtraction,  3a?^y  4- 3a?^2=a^ — b. 

Dividing   by   three   times  the   first,   a?y=~r — ,  which 
put  =^c,  ^^ 

From  this  and  the  first  equation,  we  find  as  in  example  2d. 

00= 7^ 


y= 2 

Otherwise,  dividing  the  second  equation  by  the  first, 

b 

4}b 
Hence,  4<x'^ — 4<xy+4<y^= — , 

Squaring  the  first  equation,  x^+^xy+y^^a"^. 

Subtracting  the  latter  from  the  former,  we  have 

45 — a^ 
3x''—6xy+  3y^= 5 

4<b——(i^ 
Consequently,  a?2 — 2xy+y^= — - — -, 

4:b—a^ 
And  by  extracting  the  root,  x — y=dzy/ — ^ — -, 

^     ^~3^~  «±V— -3— 

Whence,  a?= and  y= ^r 

2  ^2 

6* 


Q6  SIMPLE  EQUATIONS, 

If  we  put  -r — =c,  the  values  of  x  and  y  will  be  found 
the  same  as  before. 

^iven  ^  ^__y_^^  ^Q  £jj^  ^  g^j^jj  y  -jj  ^i^g  terms  of  s 
and  c?. 

Kesult,  a?=-^,  ^==-^ 

7.  Given  |  ^3_^,^  J^  ^^  ^^^  ^^  before. 

8.  Given  |  ^.1^2/^1,26,  to  find  a?  and  y. 

Kesult,  07=65  y=4. 
ro?— 2      10— a7_2/— 10 


9.  Given . 


4     ' 


2y+^_2^+^^+17^     Eequiredthenu- 
3  8  4  ^ 

merical  values  of  x  and  y.  Ans.  a?=7j  y=  10. 

r  a?+l:2/— 1::5:2, 

10.  Given }  ^o?      5—2/     41      2a7— 1  ,    .    .  , 

JT-"T'=12 -4—  to  find.:  and  3/. 

Kesult,  a?=4, 3/=  3. 
r  2/       4a? — 1__       4+y    a? — y 

11.  Given]  T"""!^"""  3~"^~6~' 

(  07 :  3?/ : :  4  :  7,  to  find  x  and  y. 

Kesult,  07=12, 2/= 7. 
4_    _5^__  9 

^     y'~' y~  ' 


12.   Given 


I  A+A^:^  I  A,  to  find  07  and  y. 
[  X  '^  y      0?  "•"  2 


Kesult,  07=4,  y=2. 


SmFLlS.  EQUATIONS.  67 

13.   Given  ^^'2/+^y^=  120, 

I  x^  +  y^=162.       What  are  the  values  of  x 
and  y  ?  Ans.  x=.o  or  3  .  .  2/1=3  or  5. 

14  Given  ^  x+y=12, 

I  a?^+2/^=468,  to  find  x  and  y. 

Result,  a: =7  or  5,  2/ =5  or  7. 

15  Given  ^  x'+a:y=in, 

(  ^y+y'^  =:84<,  to  find  X  and  y. 

Result,  a?=8,  y=6. 

16.*  Given  ^5^+*^='"'       .    ,  , 

I  dx-^-ey  =  ?^,  to  find  x  and  y. 

_,      ,,  em — bn         an — md 

Result,  a?=: ■ir,y= tt 

ae — do   ^      ae — do 

17    Given  ^"'*-2'- 12240, 

II.  uiven  ^  ^_^y,^  J70,  to  find  x  and  y. 

Eesult,  a;=ll,  yr=7. 

(  x^ — y^=9S,  to  determine  x  and  y. 

Result,  07=5, 2/ =3. 

19    Given  r^+*^2/=  30, 

ly.   uiven  ^  5^  ^32^^34^  to  find  x  and  y 

Result,  a:=2,  y=8. 

'^     1  1  o 

2^+-3^==^ 

-o-a^—  "^2/=  1  to  find  X  and  y . 

Result,  x=  12,  y=6. 


*  When  we  have  two  equations,  and  two  unknown  quantities, 
neither  of  which  rises  above  the  first  power,  the  required  values  may- 
be obtained,  from  the  results  of  the  16th  example,  by  simple  substi- 
tution. 


20.   Given  « 


68  SIMPLE  EQUATIONS. 

{Ix  +  ^y 

I  A        — 

21.   Given  ^ 

^       54— 8y 

2a?= — ^ —  to  find  x  and  ^. 

Eesult,  a?=:l,  2/— 6. 
0/"  arithmetical  progression, 

61.  When  the  successive  terms  of  a  series  of  numbers 
are  formed  by  the  addition  or  subtraction  of  a  constant 
number,  the  series  is  called  an  equidifFerent  one  j  or  the 
numbers  are  said  to  be  in  arithmetical  progression ;  and 
the  constant  number  is  termed  the  common  difference. 
Thus  1,  3,  5,  7,  9.  a,  azhc?,  ad=2G?,  a=l=3c?,  are  series  in 
arithmetical  progression,  the  common  differences  being  2 
and  c?,  respectively. 

In  any  series  in  arithmetical  progression,  it  is  obvious : 

1.  That  the  last  term  is  equal  to  the  first  increased  or 
diminished  by  the  common  difference  multiplied  by  the 
number  of  "terms  less  one. 

2.  That  the  sum  of  the  extremes  is  equal  to  the  sum  of 
any  other  two  which  are  equally  distant  from  them,  or  to 
twice  the  mean  when  the  number  of  terms  is  odd. 

3.  That  the  whole  sum  of  such  series  is  equal  to  the 
sum  of  the  extremes  multiplied  by  half  the  number  of 
terms. 


1.  What  is  the  sum  of  a  descending  arithmetical  series, 
of  which  the  first  term  is  10,  the  common  difference  ^,  and 
the  number  of  terms  211 

^^     20     30      20     10     ,    ^^ 
10——=— ^=— =last  term. 

/lO     10\  21     40     21     ^^.  _. 

i  — +  — ••— =-^X-^  =  140=sum  of  the  series. 

2.  Suppose  a  car  descending  an  inclined  plane  moves  5 
feet  during  the  first  second  j  15  feet  the  second  5  25  feet  the 


SIMPLE    EQUATIONS.  69 

third;  increasing  the  distance  10  feet  each  second  of  time; 
how  far  will  it  move  in  a  minute '? 

Ans.  18000  feet. 

3.  Suppose  100  loads  of  wood  placed  in  a  straight  line 
at  intervals  of  10  perches  each,  and  that  a  wagoner  is  en- 
gaged to  transport  them  to  a  place  in  the  same  line  con- 
tinued, 20  perches  from  the  nearest  pile ;  how  far  must  he 
travel  in  performing  the  service,  his  journey  being  com- 
menced at  the  place  of  deposit  1 

Ans.  160  771.  300  p. 

4.  What  is  the  sum  of  the  odd  numbers  1,  3,  5,  7,  etc., 
continued  to  n  terms  1  ,  Ans.  n^. 

Of  geometrical  progj'ession. 

62.  When  a  series  of  numbers,  are  such  that  any  term 
multiplied  by  a  constant  number,  either  integral  or  frac- 
tional, produces  the  next  following  term,  the  numbers  are 
said  to  be  in  geometrical  progression,  thus  2,  6,  18,  54 ; 
216,  144,  96,  64;  «,  «r,  ar%  ar^,  ar%  are  in  geometrical 
progression,  the  common  multipliers  being,  respectively 
3,  t  and  r. 

The  constant  multiplier  is  usually  termed  the  ratio.  In 
these  series  we  readily  perceive  that  the  last  term  is  equal 
to  the  product  of  the  first  by  the  ratio  involved  to  the 
power  denoted  by  the  number  of  terms  less  one : 

n 1. 

Thus,  the  n  term  of  the  series  «,  ar,  ar%  etc.  is  ar 

To  find  the  sum  of  a  series  in  geometrical  progression 
assume,  the  series        a  +  ar-^ar^         ar°~2+^r°~^=5,  then 

multiplying  by  r,  ar  +  ar^         ar""^ + «r°~^  +  ar° = rs^ 

whence  by  subtraction — a  4-flr"=  or   (r° — l)a=^rs — s  or 

7-° 1 

(r — l).s,  wherefore -a=s.     If  r  is  a  proper  fraction 

1 — r" 

.a=zs, 

1 — r 


70  SIMPLE  EQUATIONS. 


EXAMPLES- 

1.  What  is  the  sum  of,  2  +  6  +  18,  etc.  to  10  terms'? 

2.^g—-j)=3^«— 1  =  59048. 

2.  The  first  term  of  a  geometrical  series  being  «,  the 

m 
ratio,  the  proper  fraction  — ,  required  the  sum  of  the  series 

continued  to  infinity. 

As  —  is  a  proper  fraction,  or  n  more  than  m^  —   raised 

to  an  infinite  power,  may  he  considered  =0. 


Hence,  the  sum  required 


m     n — m 

1 

n 


2     4 

3.  Required  the  sum  of  I  +  0+9+  etc.  to  infinity. 

Result,  3. 

4.  The  series  1 — ^  +  q  —  ^^,  etc.,  being  continued  to 
infinity,  required  the  amount  %  3 

5.  What  is  the  sum  of  I+h  +  t?  continued  20  terms] 

524287 

^''''  ^524288' 

Of  harmonical  progression, 

63.  If  we  have  a  series  of  numbers  in  arithmetical  pro- 
gression, their  reciprocale,   taken  in  the  same  order,  form 

a  series  in  harmonical  progression.     Thus,  if  — ,  -y ,  — ,-y 

are  in  arithmetical  progression,  «,  5,  c,  d  are  said  to  be  in 
harmonical  progression. 


SIMPLE  EQUATIONS.  71 

Let  — .  -7-.  —  be  in  arithmetical  progression,  then  Art. 

112 

61 1 ==="T"  •*•  bc-{-ab—2ac,  or  (a-\-c)b=z2ac.     From 

the  former  of  these  equations,  be — ac=ac — ab,  whence 
arc::  b — a  :  c — b.  Hence  we  have  the  relations  of  the 
terms,  by  which  any  two  of  three  numbers  in  harmonical 
progression  being  given,  the  third  may  be  found. 

EXAMPLES. 

1.  The  first  and  third  terms  of  an  harmonical  series  are 
6  and  12,  required  the  mean. 

2X6X12     lU     ^     _,.  ,  . 

-^ --r — =-— -=8.    1  he  number  souo;ht. 

6+12         18  ° 

2.  The  first  and  second  terms  of  an  harmonical  series 
are  12  and  16  ,  what  is  the  third '?  Ans.  24. 


SIMPLE  EQUATIONS. 

Containing  three  or  more  unknown  quantities. 

6'!'.  The  unknown  quantities  may  be  eliminated  one 
after  another,  by  the  first  three  rules  laid  down  in  the 
last  section.  Or  multiply  each  given  equation,  except  one, 
by  an  indefinite  quantity  5  add  the  remaining  given  equa- 
tion, and  all  the  new  equations  together;  and  assume  the 
sum  of  the  co-efficients  of  each  required  quantity,  except 
one, equal  to  0;  whence,  as  many  equations  as  there  are 
assumed  multipliers  will  be  formed;  and  thence  the  values 
of  those  multipliers  may  be  determined,  and  consequently, 
the  last  unknown  quantity. 

There  are  other  methods  which  may  often  be  advan- 
tageously applied,  but  which  can  be  learned  only  from 
practice. 


72  SIMPLE  EQUATIONS. 


EXAMPLES. 


(00+  y+  2r=29,         >j 

I  0^+22/ +  32^=62,  to    find  the   values    of 

Given      a:       1         1  [o^,  2/,  and  z. 

2         1 
Multiply  the  last  equation  by  2,  a?+-^y-i--^2;=20. 

Subtract  this  from  the  first  equation,  — y+— 2;=9. 

3 

Multiply  by  3,  and  y+—z=2'7. 

Subtract   the   first    equation  from    the  second,   and 

2/  +  2z=33.     {A,) 
From   this   equation  subtract  the  former,  and  we  find 

-2;z=6^  or  2;=  12. 

Take  double  the  last  equation  from  equation  A^  and  3/= 9. 
Hence  y+2:=21,  and  this  equation  subtracted  from  the 
first  leaves  a? =8. 

Otherwise, — From  the  given  equations, 

a? =29 — y — z, 

a?=62— 2?/— 3;^. 

on      ^        1 

Whence  29— y— 2^=62—23/— 32?. 

2        1 

And  20— —  2/— -2r=:29— y— 2r. 

From  the  former  of  these,  y=33 — 22:. 

3 

And  from  the  latter,  y=27 —z. 


SIMPLE  EQUATIONS.  73 

Therefore,  'jll—~z=33—2z, 
/u 

Or,  54—3^=66—4^^;  r,z=12.'^ 

Buty=33—2z=9. 

And  x=29 — y—z=:8. 

Or  multiplying  the  second  and  third  equations  by  m  and 
n  respectively,  mx-{-2my-i-3mz=62m,, 

■^nx-\--ny+-nz=10n. 

Hence,  by  adding  these  equations  to  the  first, 

x-{-mx-\—^nx-\-y-{'2my-\-—ny-\-z  +  3mz-{—j-nz=: 
29 +  62m +1071. 

1  1 

Assume  l+2;?^  +  -?^=:0,  and  l^-3;?^4--7^=0 
o  4* 

3  1 

From  the  former  of  these  equations  -  +  37?^4--7^=rO. 

-^  Z 

Hence  m= — -. 
o 

29-^-20 
29  +  62m  +  -i0n     ^^      6 

Whence,  a:= ^j =- 


1  1  '• 

l+m  +  —n         1— 1 

Again    making  l+7?z+— 7i=0,  and  l4-37?^+— 7i=0, 
We  have  n= —.B.ndm= — p-. 

0  0 


*The  sign  .«.  is  used  in  place  of  the  word  therefore. 
7 


74«  SIMPLE    EQUATIONS. 

29+62m+10n      ^ 
Consequently,  y= —  =9, 

Hence,  2:=  12. 

2.  Given]  a7+22/+32r=  105,   , 

(  07+3^+42;=  134,  to  find  a?,  y,  and  z. 

Subtract  the  first  from  the  second,  y-{-2z=^2. 
Second  from  third,  ?/  +  2:=29. 
Subtracting  the  last,  2r=23. 
r.y=:6. 

And  a?=24. 

C  x+y=20, 

3.  Given  <  a?-f-2r=24, 

^  y+2:=30,  to  find  0?,  y,  and  z. 

Eesult,a?=7,  ^=13,  0=17. 

C  7a?+53/+22:=79, 

4.  Given?  8a7-|-7i/ +9^=122, 

(    a?+4y  +  52:=55,  to  determine  the  values  of 
a?,'y,  and  z.  Result,  a?=4,  y=9,  z=3, 

(  ^— y=2, 

5.  Given  <  a? — z=3, 

(  2/  +  2r=9,  to  find  a?,  3/,  and  z. 

Result,  07=7,  y=5,  z=4f. 

-  C  2a:+3y+4;^=38, 

6.  Given  <  5a?+ 72/4- 112:= 98, 

(  7a? -f  9y  + 152:=  132.  to  find  a?,  y^  and  z. 

Result,  07=3,  2/ =4,  z=5 

C  aa7+5^+C2r=77?, 

7.  Given  <  dx+ey  +fz = w, 

(  gX'\'hy-\-kz=p^  to  find  a?,  y,  and  z. 


SIMPLE  EQUATIONS. 


75 


Kesult,* 


8.    Given 


9.   Given  < 


_  kem — hfm  -f  hen — hkn + hfp — ecp 
aek — ahf-\-  dhc — dbk-\-gbf — gee 
akn — afp  -f-  dcp — dkm  -f-  gfm — gen 
^  ~~  aek — ahf-\-  dhe — dbk  +gbf—gec 

aep — ahn  +  d/im — dbp  -\-gbn — gem 

~  aek — afif-\-dhc — dbk  +gbf' — gee 

,^4-y+2:,  =  33,  to  find  z?,  a:,  y,  and  z. 

Result,  ?;=9,  a?=10,  y=ll,  2:=  12. 
f4a7+3y+-2:     22/+22r— a7+l_       a?— z— 5 
10  15  ''        5         ' 

9a?  +  5y— 22f     2a?  +  y— 32r_7y+2r  +  3  .  1 

12  5     -    n 


+  6^ 


5y4-32r     2a7+3y— 2r 


+  22r=y— 1  + 


3a:+2y4.7 


4  12 

Required  the  values  of  a:,  y,  and  2^. 

Ans.  07=9,  y=7,  2?=  3, 

f  3v  +  4a?-f-5y-h62r=:102, 

(  62;-!- 5a: — 7^  +  2^=7,  to  find  v,  a?,  y,  and  z. 

Result,  2;= 5,  a?=4,  y=7,  2^=6. 

Examples  producing  simple  equations. 

Required  to  find  two  numbers  whose  difference  shall  be 
4,  and  the  difference  of  the  squares  64. 

Let  07  =  the  greater  number,  and  y  =  the  less. 

Then,  by  the  question,  |  ^^^^Jq^^ 
Dividing  the  2d  equation  by  the  1st,  a?+y=16 


*  From  these  general  results,  the  values  of  x,  y,  z^  in  the  preceding 
examples,  may  be  found  by  simple  substitution. 


76  SIMPLE  EQUATIONS. 

Taking  the  sum  and  difference  of  this  equation  and  the 
first,  2x=20,  and  2y=12. 

.•.  a?=10,  and  y=6. 
Otherwise.   Let  x  =  the  less  number,  then  x+4}  =  the 
greater : 

Whence,  by  the  question,  a7-|-4]^ — a:^=64<. 

Or  807+16  =  64. 
.'.  by  transposition  and  division,  x=6,  and  a:-h4=10. 

2.  A  person  having  bought  three  loads  of  grain,  the 
first,  consisting  of  30  bushels  of  rye,  20  of  barley,  and  10 
of  wheat,  for  230  francs  ;  the  second,  containing  15  bushels 
of  rye,  6  of  barley,  and  12  of  wheat,  for  138  francs;  the 
third  consisting  of  10  bushels  of  rye,  5  of  barley,  and  4  of 
wheat,  for  75  francs ;  required  the  price  per  bushel  of  each 
of  these  kinds  of  grain. 

C  a  bushel  of  rye  cost  a?,        1 
Suppose  <  a  bushel  of  barley,  y,  >  francs 

(  and  a  bushel  of  wheat,  z,  ) 

C  3007+203^+ 10^=230,(^.) 

Then  by  the  question,  }  1507+6^+122;=  138, 

(  10o7+5y+42;=75. 

Multiplying  the  2d  equation  by  2,  and  the  3d  by  3, 

30o7+122/+24;^=276, 

3007+151/  + 12^=225,  (B,) 

Taking  the  difference  of  these  equations, 

32/— 12z=— 51.  (C.) 

Take  the  difference  of  equations  (^d,)  and  (B,)  and 

5y— 22;=5. 

Multiplying  this  by  6,  30y—12z=30. 

And  subtracting  equation  (C) 

27y=81.     Whence,  y=  3. 

15__5                      23—6—5     , 
.•.  z= — - —  =  5,  and  07= =4. 


SIMPLE  EQUATIONS.  77 

3.  Required  to  find  two  numbers,  such  that  their  dif- 
ference multiplied  by  their  product,  shall  be  308,  and  the 
difference  of  their  cubes  988. 

Let  X  —  the  greater,  and  y  =  the  less. 
Then  from  the  question, 

(x — y).a?y,  or  x^y — xy^=30S, 
And  0?^— i/3^988. 

From  the  latter  taking  thrice  the  former, 

We  have  x'^ — 3x''-y  +  3xy'^ — 2/3=64. 
And  taking  the  cube  root,  x — y—4<,  (^A,) 

Dividing  the  sum  of  the  two  primary  equations  by  this 
last, 

x^  -f  x'^y — j?y^ — y^ 

=a?2-f2a?2(+2/2=324. 

Hence,  by  extracting  the  square  root,  a7+y=18.  (J5.) 
Taking  the  half  sum  and  half  difference  of  equations 
{A.)  and  (jB.) 

18+4  18-— 4 

07=— ^=11,  and  2/=— ^=7. 

4.  The  sum  of  two  numbers  multiplied  by  their  differ- 
ence is  20,  and  the  difference  of  their  4th  powers  1040 ; 
what  are  the  numbers  1 

Let  X  =  the  greater,  y  =  the  less. 

Then  from  the  data,  \  ^y^y^irolol 

Dividing  the  latter  by  the  former,  x'^-\-y'^=52. 
Taking  the  sum  and  difference  of  the  first  and  last  equa- 
tions, 2a?2=72,  and  2y^=32. 

.'.  x=VS6  =  6. 
And  y=  ^16=4. 

5.  The  difference  of  two  numbers  is  5,  and  the  differ- 
ence of  their  cubes  1685  ;  what  are  the  numbers '? 

Let  X  =  the  greater,  y  =  the  less. 
7* 


78  SIMPLE  EQUATIONS. 

Then,  per   dataA^T''^^'7^\r,oK 
'  -r  J  ^  ^> — y.—  X685. 

By  division,  ^;;;:::^=a?2 4- ^y+S''^  337,  {A.) 

Squaring  the  first  equation,  a?^ — 2a:y4-2/2=25. 

By  subtraction,  3a?2/=312. 

Adding  i  of  this  equation  to  equation  (.^.) 

Extracting  the  square  root,  07+3/=  21. 

From  this  and  the  first  equation,  a?=:13,  and  2/~8. 

Or  from  the  second  equation  subtract  the  cube  of  the 
first,  whence  So^^y — 30:3/2=  1560;  and  dividing  this  equa- 
tion by  three  times  the  first,  ccy=  104' ;  v^rhence  as  in  page  63, 


^25+416  +  5                         ^25  +  416— 5 
a?= o =13.  and  y— ^ =  S* 

6.  Required  to  find  two  numbers,  such  that  half  the  first 
added  to  a  third  of  the  second  shall  make  9,  and  a  fourth  of 
the  first,  added  to  a  fifth  of  the  second,  shall  make  5 

Ans.  8  and  15. 

7.  A  testator  bequeaths  to  his  widow  §  of  his  estate,  to 
his  son  i,  and  the  remainder,  which  is  found  to  be  756  dol- 
lars, to  his  daughter;  what  was  the  estate  left] 

Ans.  2835  dollars. 

8.  There  are  two  numbers  in  the  ratio  of  3  to  5,  and 
their  sum  is  one  fourth  of  the  difference  of  their  squares; 
what  are  the  numbers'?  Ans.  6  and  10. 

9.  The  ages  of  a  man  and  his  wife,  at  the  time  of  mar- 
riage, were  in  the  ratio  of  7  to  6,  and  at  the  end  of  30 
years,  they  are  found  to  be  as  17  to  16.  Quere  their  ages 
at  the  former  period  1  Ans,  21  and  18. 

10.  A  post  being  i  of  its  length  in  the  earth,  i  in  the 
water,  and  10  feet  above  the  water;  what  was  its  whole 
length  1  Ans.  24  i^^L 


SIMPLE    EQUATIONS.  79 

11.  Divide  1100  dollars  among  A.  B.  and  C,  so  tha4 
B.  shall  have  100  dollars  more  than  A.,  and  C.  150  more 
than  B.J  what  is  the  share  of  each'? 

Ans.  A.  250,  B.  350,  C.  500. 

12.  There  are  21  persons,  whose  ages  form  an  equi-dif- 
ferent  series ;  the  age  of  the  eldest  is  4  times  that  of  the 
youngest,  and  the  sum  of  all  their  ages  is  525  years  5  how 
old  is  the  youngest  1  Ans.  10  years. 

13.  At  a  certain  election,  1296  persons  voted,  and  the 
successful  candidate  had  a  majority  of  120 ;  how  many 
voted  for  eachl 

Ans.  708  for  one,  and  588  for  the  other. 

14.  A  servant  agreed  to  serve  his  master  for  ^8  a  year 
and  his  livery,  but  was  turned  away  at  the  end  of  7  months, 
and  received  only  £2  13s.  M.  besides  has  livery;  wlmt  is    . 
the  price  of  his  livery'?  Ans.  £4}  16s, 

15.  A  servant  having  eloped  from  his  master,  travels  14? 
hours  in  the  day,  at  the  rate  of  3^  miles  an  hour;  at  the  ^ 
end  of  two  days,  a  courier  is  sent  in  pursuit,  who  rides  9 
hours  in  the  day,  at  the  rate  of  7  miles  an  hour;  in  what 
time,  and  at  what  distance  will  he  overtake  him'? 

Ans.  7  days,  and  441  miles. 

16.  The  sun's  mean  daily  motion  in  the  ecliptic  is  59'  8", 
and  that  of  the  moon  13^  10'  35";  what  is  the  time  of  a 
mean  synodic  revolution  of  the  moon,  viz.  a  revolution  from 
conjunction  with  the  sun  to  conjunction  again '? 

Ans.  29  days,  12  hours,  44  minutes  nearly. 

17.  A  farmer,  having  hired  a  laborer,  on  condition,  that 
for  every  day  he  wrought  he  should  receive  50  cents,  and 
for  every  day  he  was  idle  he  should  forfeit  20  cents,  finds 
at  the  end  of  420  days,  that  neither  is  indebted  to  the 
other;  how  many  days  did  he  labor.  Ans.  120. 

18.  A  certain  number,  consisting  of  two  places  of  figures, 
is  equal  to  the  difference  of  the  squares  of  its  digits ;  and  if  . 
36  be  added  to  it,  the  order  of  the  digits  will  be  inverted; 
what  is  the  number  1  Ans.  48.^ 

19.  A  vintner  proposing  to  mix  three  softs  of  wine,  viz. 


80  SIMPLE  EQUATIONS. 

at  65  cents  per  gallon,  at  45  cents,  and  at  35  cents;  the 
number  of  gallons  at  65  cents,  to  the  number  at  35  cents, 
being  as  5  to  3,  so  as  to  compose  a  hogshead  worth  50 
cents  per  gallon.     Quere  the  number  of  each  1 

Ans.  22iat  65,  27  at  45,  and  13iat  35. 

20.  There  are  4  equi-difFerent  numbers,  whose  sum  is  56, 
and  the  sum  of  whose  squares  is  864 ;  what  are  the  numbers'! 

Ans.  8,  12,  16,  and  20. 

21.  If  38  federal  dollars  and  11  English  crowns  be 
given  for  271  French  francs,  and  76  dollars  and  33  crowns 
for  610  francs;  at  how  many  francs  were  the  dollar  and 
crown  respectively  estimated '? 

Ans.  the  dollar  5||,  the  crown  6f-y. 

22.  There  is  a  number  consisting  of  three  digits  in  arith- 
metical progression,  whose  sum  is  21 ;  and  if  to  the  number 
396  be  added,  the  sum  will  be  expressed  by  the  same  digits 
in  an  inverted  order;  what  is  the  number  1      Ans.  579. 

23.  Eequired  to  find  four  numbers,  such  that  the  con- 
tinued product  of  the  1st,  2d,  and  3d  shall  be  24 ;  the  pro- 
duct of  the  1st,  2d,  and  4th,  30 ;  of  the  1st,  3d,  and  4th, 
40 ;  and  of  the  2d,  3d,  and  4th,  60.      Eesult,  2,  3,  4,  5. 

/  24.  It  is  required  to  find  three  numbers,  such  that  the 
first  multiplied  by  the'  sum  of  the  other  two  shall  be  96  ; 
the  second  multiplied  by  the  sum  of  the  other  two  shall  be 
105  ;  and  the  third  multiplied  by  the  sum  of  the  former  two 
shall  be  117.  The  numbers  are  6,  7,  and  9. 

25.  What  fraction  is  that,  to  the  numerator  of  which,  if 
1  be  added,  the  value  will  be  |,  but  if  to  the  denominator 
1  be  added,  the  value  will  be  i  1  Ans.  y4_. 

26.  A  person  bought  a  chaise,  horse,  and  harness,  for 
150  dollars;  the  price  of  the  chaise  was  twice  the  price 
of  the  harness,  and  the  price  of  the  horse  as  much  and 
half  as  much  as  the  price  of  the  chaise  and  harness ;  what 
was  the  cost  of  each  % 

Ans.  harness  20  dollars,  chaise  40,  horse  90. 

27.  Two  persons,  A.  and  B.,  have  the  same  income;  A. 
saves  ^  of  his  yearly ;  but  B.,  by  spending  as  much  in  3 


SIMPLE  EQUATIONS.  81 

years  as  A.  does  in  4,  finds  himself,  at  the  end  of  ^ye  years, 
200  dollars  in  debt  5  what  was  the  income  1 

Ans.  600  dollars. 

V  28.  A.  and  B.  put  equal  sums  in  trade:  A.  gained  a  sum 
equal  to  i  of  his  stock,  and  B.  lost  450  dollars;  when  A.'s 
money  was  double  of  B's;  what  was  the  sum  laid  out  by 
each  1  Ans.  1200  dollars. 

29.  Two  persons  comparing  their  ages,  find  them,  at 
present,  in  the  ratio  of  7  to  5,  but  that  30  years  ago,  they 
were  in  the  ratio  of  2  to  1 ;  what  are  their  ages '? 

Ans.  70  and  50. 

^  ^^  30.  A.  and  B.  began  trade  with  equal  sums  of  money. 
In  the  first  year  A.  gained  J240,  and  B.  lost  .£40;  but  in 
the  second,  A.  lost  one  third  of  what  he  then  had,  and  B. 
gained  a  sum  which  was  £40  less  than  twice  what  A.  had 
lost ;  when  it  appeared  that  B.  had  twice  as  much  as  A. ; 
what  sum  had  each  of  them  at  first  1  Ans.  £320. 

^  ^  31.  A  farmer  sold  96  loads  of  hay  to  two  persons.  To 
the  first  one  half,  and  to  the  second  one  fourth  of  what  his 
stack  contained ;  how  many  loads  were  in  the  stack  1 

Ans.  128. 

32.  If  116  be  divided  into  four  parts,  in  such  manner, 
that  the  first  being  increased  by  5,  the  second  diminished 
by  4,  the  third  multiplied  by  3,  and  the  fourth  divided  by 
2,  the  results  will  all  be  the  same ;  what  are  the  parts  % 

Ans.  22,  31,  9,  and  54. 

/•^  v^  33.  A  shepherd,  in  time  of  war,  was  plundered  by  a 
party,  who  took  from  him  i  of  his  flock,  and  i  of  a  sheep 
more  ;  another  party  took  from  him  ^  of  what  he  had  left, 
and  ^  of  a  sheep  more ;  afterward,  a  third  party  took  \  of 
what  remained,  and  \  a  sheep  more,  when  he  had  but  25 
left ;  how  many  had  he  at  first  \  Ans.  103. 

J  34.  A  person  has  two  horses  and  a  gig ;  the  gig  is  worth 
150  dollars.  When  the  first  horse  is  attached  to  the  gig, 
the  value  of  the  two  is  twice  that  of  the  second  ;  but  when 
the  second  horse  is  put  to  the  gig,  the  value  is  three  times 


82  SIMPLE  EQUATIONS. 

that  of  the  first  horse  ;  what  were  the  horses  respectively 
worth  ]  Ans.  the  first  90  dollars,  the  second  120. 

/  35.  When  a  company  at  a  tavern  came  to  pay  their 
^  reckoning,  they  found,  that  if  there  had  been  three  persons 
more,  they  would  have  had  a  shilling  a  piece  less  to  pay ; 
but  if  there  had  been  two  less,  they  would  have  had  a  shil- 
ling a  piece  more  to  pay  5  required  the  number  of.  persons, 
and  the  quota  of  each  1 

Ans.  12  persons  5  quota  of  each  5s. 
36.  There  are  three  equi-different  numbers,  whose  sum 
is  324,  and  the  first  is  to  the  third  as  5  to  7 ;  what  are  the 
numbers'?  Ans.  90,  108,  and  126. 

J  37.  A  man  and  his  wife  usually  drank  a  cask  of  beer  in 
12  days ;  but  when  the  man  was  from  home,  it  lasted  the 
woman  30  days ;  how  long  would  the  man  alone  be  in 
drinking  it  1  Ans.  20  days. 

J  38.  A.  and  B.  can  perform  a  piece  of  work  in  8  days,  A. 
and  C.  in  9  days,  and  B.  and  C.  in  10  days ;  how  many 
days  would  they  severally  require  to  perform  the  same 
work'?  Ans.  A.  14||  days,  B.  17|f,  and  C.  233^. 

/       39.  The  hypothenuse  of  a  right  angled  triangle  is  13, 
"^   and  the  area  30  5  what  are  the  other  two  sides  1  * 

Ans.  12  and  5. 
40.  There  are  four  numbers  in  geometrical  progression, 
the  sum  of  the  extremes  is  18,  and  the  sum  of  the  means 
12  j  what  are  the  numbers  if         Ans.  2,  4,  8,  and  16. 

*  To  solve  this  problem,  it  must  be  recollected  that  the  square  of 
the  hypothenuse  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides,  and  that  the  area  is  half  the  product  of  those  sides. 

t  To  solve  this  question  by  simple  equations,  put  x,  y  to  denote  the 

means,  then  since,  when  three  numbers  are  in  geometrical  progression, 

the  square  of  the  mean  is  equal  to  the  product  of  the  extremes,  the  ex- 

x^  yn 

tremes  will  be  expressed  by  — ,   and  — .       Hence   rr4-y=12,  and 

— [-— =1^'    From  the  last  ics-f-j^s^riSariy.     Subtracting  this  equa- 
y      ic 

tion  from  the   cube   of  the  first,  3rr2y-}-3a;y3  _  1728 — ISxy,     But 

2x^  H-  3a;y3==3(a;-f-y).  xy  =  ^Qxy.     Hence  5^xy  =  1728  or  ir^=32. 

Whence  x  and  y  are  found  as  in  page  64. 


# 


SIMPLE  EQUATIONS.  83 

41.  Required  to  find  four  numbers  in  geometrical  pro- 
gression, such  that  the  difference  of  the  means  shall  be  100, 
and  the  difference  of  the  extremes  620. 

The  numbers  are  5',  25,  125,  and  625, 

42.  There  are  three  numbers  in  harmonical  progression, 
whose  sum  is  26,  and  the  difference  between  the  second,  and 
third  exceeds  the  difference  between  the  first  and  second 
by  2  5  what  are  the  numbers  %  Ans.  6,  8,  and  12. 

43.  A  person  having  £21  6s.  sterling  in  guinea^/- and 

1/   crowns,  pays  a  debt  of  il4  175.,  and  then  finds  the  Humber*.''  ^ 
of  guineas  left  equal  to  the  number  oi*^crownstpaid  awS,y; 
and  the  number  of  crowns  left  equal  to  the  number  of 
guineas  paid ,  how  many  of  each  had  he  at  first  1 

Ans.  21  of  each. 

^  44.  There  is  a  number  consisting  of  two  digits,  which  is 
equal  to  four  times  the  sum  of  those  digits  j  but  18  being 
added,  the  order  of  digits  will  be  inverted.  Quere  the 
number  1  Ans.  24. 

45.  Required  to  find  three  such  numbers,  that  i  of  the 
first,  i  of  the  second,  and  i  of  the  third  added  together 
shall  make  46  ;  i  of  the  first,  i  of  the  second,  and  J  of  the 
third  shall  make  35  ;  and  i  of  the  first,  |  of  the  second,  and 
1  of  the  third  shall  make  28^.       Result,  12,  60,  and  80. 

46.  A  person  having  £22  14^.  sterling  in  crowns,  gui- 
V  neas  and  moidores,  finds  that  if  he  had  as  many  guineas  as 

he  has  crowns,  and  as  many  crowns  as  he  has  guineas,  the 
whole  sum  would  be  ^36  6s, ;  but  if  he  had  as  many  moi- 
dores as  he  has  crowns,  and  as  many  crowns  as  he  has 
moidores,  his  money  would  amount  to  JG45  16s,  How 
many  had  he  of  each  1 

Ans.  26  crowns,  9  guineas,  and  5  moidores. 
.      47.  Required  two  numbers,  such  that  their  sum,  differ- 
ence, and  product  may  be  as  the  numbers  3,  2  and  5  re- 
spectively. Result,  10  and  2. 

48.  The  sum  of  the  first  and  third  terms  of  four  num- 
bers in  geometrical  progression  is  74,  and  the  sum  of  the 
second  and  fourth  444 ;  what  are  the  numbers '? 

Ans.  2,  12,  72,  and  432. 


€^^' 


84  QUADRATIC  EQUATIONS. 

^  49.  A.  and  B.  enter  into  trade  with  different  sums,  A. 
gains  750  dollars,  and  B.  loses  250  dollars;  when  their 
stocks  are  found  to  he  as  3  to  2.  But  if  A.  had  lost  250 
dollars,  and  B.  had  gained  500  dollars,  their  stocks  would 
have  been  as  5  to  9.  What  was  the  original  stock  of  each  \ 
Ans.  A.  1500  dollars,  B.  1750  dollars. 


Section  VII. 

QUADEATIC  EQUATIONS. 

65.  When  an  equation  contains  the  square  and  simple 
power  of  an  unknown  quantity ;  or  in  general,  two  powers, 
one  of  w^hose  indices  is  double  the  other,  of  an  unknown 
quantity,  whether  that  unknown  quantity  be  simple  or 
compound,  it  is  called  an  adjected  quadratic  equation. 

Rule  I. 

Arrange  the  terms,  by  transposition  or  otherwise,  so  that 
the  highest  power  of  the  unknown  quantity  shall  be  con- 
tained in  the  first,  and  the  other  power  in  the  second  term 
on  the  left  side  of  the  equation  ;  and  the  term  or  terms  con- 
sisting of  known  quantities,  on  the  other  side. 

When  the  highest  power  of  the  unknown  quantity  con- 
tains a  co-efficient,  divide  the  equation  by  that  co-efficient. 
Then  add,  tp  each  member  of'  the  equation,  the  square  of 
half  the  co-efficient  of  the  unknown  quantity,  (or  the  nu- 
merical part  of  the  second  term,)  and  the  first  member  of 
the  equation  will  be  a  complete  square. 

Extract  the  root  and  find  the  value  of  the  unknown 
quantity  by  the  former  methods. 

Rule  2. 

^^,  Arrange  the  terms  as  before.  When  the  highest 
power  of  the  unknown  quantity  has  a  co-efficient,  multiply 
the  equation  by  four  times  that  co-efficient,  and  to  each 


QUADRATIC  EQUATIONS.  85 

member  of  the  resulting  equation,  add  the  square  of  the  co- 
efficient of  the  unknown  quantity  contained  in  the  primary- 
equation  5  then  the  first  member  of  the  resulting  equation 
will  be  a  complete  square. 

Extract  the  root,  and  proceed  by  the  former  methods, 
to  determine  the  value  of  the  quantity  sought. 

EXAMPLES. 

Given  2x^ — 9a?=266,  to  find  the  value  of  a?. 

9 
By  rule  1. — ^Dividing  by  2,  a?^—  -x— 133. 

9\2  ,  ,  9       81     2209 


Adding  V-)    to  each  member,  x^ — -o?^     _ 

9     47 
Extracting  the  root,  x--:=~ 

4r         4b 

.-.  a?=14. 
By  rule  2.— Multiplying  by  8,  16a?2-^-72a7=2128. 
Adding  81  to  each  member,  160?^— 72a? +8 1=2209 
Extracting  the  square  root,  4a: — 9=47. 

r,x= — z —  =  14,  as  before. 

4 

2.  Given  ax^+bx=c,  to  find  the  value  of  a?. 

b        c 
By  rule  1. — Dividing  by  a,  x^-\--x—-» 

52  ^  b        ¥     b^-h4<ac 

Adding^,  to  each,  c^+^+-=-^- 


By  evolution,  0;+-=—^—- 


By  transposition,  a?= ^ 

By  rule  % — Multiplying  by  4cj,  ^a'^x^+^ahx^iac. 
Adding  5^  to  each,  ^a''x^-\-^abx-\'h'^^^ac-^h^. 
8 


86  QUADRATIC    EQUATIONS. 


By  evolution,  2ax-\-b=^y/4^ac-\-b^, 

•D     ^              '.'           ;i  J-   •  •                VUc  +  b^'—b 
J3y  transposition  and  division,  x= 

3.  Given  20?""  +  3a?''  =  2,  to  find  the  value  of  x. 

#     3  - 
Byrule  1.     ^  +x^'  =  l. 

r  .1.  f     3  A      9      25 

Comp.  the  sqr.  x  -\-^x  +15=15 

-     3     5 

By  evolution,   a7^+j=j 

^_5— 3__1 

•'•^  ■"~T~~2 

Whence,  a?=:(^)  =g 

By  rule  2.     16  a?"  +24a?' =  16. 
Adding  9  to  each,  16  x^  +24>x  +9  =  25. 


By  evolution,  4  a:" +3=5. 

"2 


i     1 
By  transp.  and  division,  x  =-^ 


Therefore.     oc=- 
o 

4.  Given  a:* — 12a?3+44a?2— 48a?=9009,  to  determine  x. 
This  equation  is  equivalent  to  to  the  following: 

a?*—12a:3+36a?'2+8a?2— 48^=9009, 
Or  to  this,  (a?2— 6a?)2+8.(a?2_6a?)=9009. 
In  which,  x^ — 6x  may  be  considered  as  the  unknown 
quantity. 

Hence,  proceeding  as  in  the  former  examples, 


QUADRATIC  EQUATIONS.  87 

(a^3__6a?)2+8.(a?2— 6^)  + 16=9025. 
.'.x^ — 6a?4-4  =  95,  or  a:^ — 6a7=91. 
Whence,  x^—6x  +  9  =  100. 

...  0?— 3=10,  or  07=13. 

5.  Given  ^(x — -)  +  v^  (  1 )=^)  to  find  the  value 


of  X, 


Multiplying  by  \/a?,  ^/x"^ — l  +  y/x — l=Xs/x, 

bquaring  this,  x'^ — 1  +  2  \/  ^^ — ^^ — ^  + 1  +^ — 1  =  ^^• 
By  transposing  and  changing  the  signs, 

a?^ — x'^ — X  + 1 — 2  \/  0?^ — X" — X  + 1 = — 1 . 


Here  y/x^ — a?^ — ^  +  1?   niay  be  considered  as  the  un- 
known quantity. 

Hence,  adding  1  to  each  side, 


0.3 — ^2 — a?  4- 1 — 2  s/  a?3 — x'^—x  4-1  +  1  =  0. 


...  ^x'l — x'2 — ^4-1 — 1=0,  or  v/a:^ — a:^ — a? +1=1. 

.^x^—^x^ — a?+l=l; 
Whence  a?^ — a:=l. 
15  1         5     1^ 

1     1     , 

•••^=2+2^ 
6.  Given  3a^2"— 2a7"+3=ll,  to  find  x. 

Sa?'^"— 2a?"=8. 

2         8 
By  rick   1.     a?^" — -a7"=  - 

2         1      25 

^       3^  +9"  9 

0?"—^=^,  or  a? =2". 


88  QUADRATIC     EQUATIONS 

By  rule  2.     36a?'2° — 24a?°=96. 
36j:3°— 24a7°  +  4=100. 
.6a?°— 2=10. 

12 

a?°=-7r-=2,  or  07=  V  2,  as  before. 

7.  Given  a?*°— 2a:3'^  +  a?°=132,  to  find  the  value  of  a? 
This  equation  may  be  changed  to 

a?4°_2a?^°  4_a;2°_a72n  +  37^==  1 32. 
Which  is  (o?^"— a?")^— (a?^"— a?°)=132. 
Complete  the  square, 

1  529 

(a?2°_a?°)'2— (a?3°— 0?")+-=  132i=^ 

1     23 

Extract  the  root,  a?^° — a?° — o=ir 

/O  At 

.-.  a?2»— a?''=12. 

1  49 

Complete  the  square,  a?^° — 0?° +^=12i=-^ 

1     7 
Extract  the  root,  x"" — 9  =  9 

.«.  a?°=4,  or  07=  V 4. 

8.  Given  a?^4-12a7=64.      Kequired  the  value  of  a?. 

Ans  07=4. 

9.  Given  7o?2+5o7  +  4=82,  to  find  o?.  Ans.  07=  3. 

10.  Given^o?^— -07+71  =  8,  to  find  07. 

Ans.  07=14 

11.  Given  o?H20o7^=224,  to  find  07.  Ans.  07=  2. 

12.  Given  507^ — 4o7+3=159,  to  find  07.        Ans.  07=6. 

35 3^ 

13.  Given  607 -| =44,  to  find  07. 

Ans.  07=7,  or  |.* 

*  This  example  falls  under  the  ambiguous  case,  which  is  explained 
in  the  next  note. 


QUADaATIC  EQUATIONS.  89 

14.  Given  4-0? -—=14,  to  find  the  value  of  a:. 

Ans.  x=4>, 

1221 4^x 

15.  Given  3a? =2,  to  find  the  value  of  a?. 

X  ' 

Ans.  07=19. 

.r      r.'  n  ^O?— 3       ^  3x—6  ^      , 

16.  Given  5x ^=^oc-i ^— -  to  find  x. 

X — o  z     ' 

Ans.  a?=4. 
67.  The  foregoing  solutions  have  been  effected  by  exe- 
cuting the  whole  process  in  each  case  \  but  it  is  more  con- 
venient in  practice,  as  v^ell  as  more  elegant,  to  solve  all 
such  equations  by  a  few  general  formidcR.  In  order  to 
which  it  is  to  be  observed,  that  all  adfected  quadratic 
equations  are  reducible  to  one  of  the  following  forms. 

2.  x''—1ax=b,       C  (^.) 

3.  x" — 2ax=^ — h,  ) 

These  equations,  when  the  squares  are  completed,  be- 
come, 

1.  a?^-}-2aa?+a2_.^3_j_^ 

2.  0?^ — 2ax  +  a^=a^+b 

3.  a:^ — 2ax-i-a^=a^ — b. 

Hence,  by  evolution  and  transposition,  we  have 

1.  x= — a-\"^a^-\-b. 

2.  x=a'\'^/a^-rb, 

3.  x=a±is/  a^—bJ^ 

Adfected  quadratic  equations  are  sometimes  expressed 
by  one  of  the  following  forms. 

1.  o^x'^-\-bx=^c,       ^ 

2.  ax^'-bx=c.       >(B.) 


3.  ax''- 


See  note,  page  90. 
8* 


90  QUADRATIC  EQUATIONS. 

Multiplying  these  equations  severally  by  4a,  and  adding 
b^  to  each  member,  they  become 

1.  4<a''x^+4!abx+b^=b^+4^ac. 

2.  4a V — ^abx+b'^=b'^+4<ac: 

3.  4a2a?2 — ^^abx+b^^b'' — 4ac. 

The  roots  being  extracted,  the  values  of  a?  are  easily 
found. 


—b+^b^+4^ac 

1.  x==. 

2a 


b+s/b^+4<ac 
^-  ^= 2a ■ 


bzt:s/b^—4<ac* 
3.  x= 

2a 

From  these  formulce^  any  adfected  quadratic  equation 
may  be  resolved,  by  assuming  for  a  and  5,  in  the  equations 
(j1,)  and  for  a,  b,  and  c,  in  the  equations  (5,)  such  values 
as  will  render  the  general  equation  identical  with  the  par- 
ticular one  under  consideration,  and  substituting  the  values 
thus  assumed  in  the  final  equation. 

Simple  quadratic  equations  are  evidently  solvable  by  the 
SQ,me  formulcB  if  we  assume  the  co-efficient  in  the  second 
term  =0. 

♦The  value  of  Xj  in  the  third  case,  is  ambiguous;  because, 
Vx^ — 2ax-{-az=x — a^  or  a — x,  and  \/ 4a^x^ — 4abx'{-b^—2ax — 5,  or 
b — 2ax  ;  and  there  is  nothing  in  the  general  equation  to  show  whether 

X  is  greater  or  less  than  a  in  the  former  case,  or  than  — -  in  the 
latter.  The  ambiguity  does  not  exist  in  the  second  case,  because 
V  a^  +  b  must  be  greater  than  «,  and  y/b2-\'4ac  greater  than  b,  and 
therefore,  cannot  be  equal  respectively  to  a — .r,  and  b — 2aXf  unless  x 
be  negative.  When  the  sign  of  x  is  doubtful,  the  first  and  second 
cases  are  also  ambiguous. 


QUADRATIC    EQUATIONS.  91 

17.  Given  9a?3— 7^^=116. 

Dividing  by  9,  x^ — -a?=-^ 

Comparing  this  equation  with  the  2d  form,  (./^,)  we  have 

2a=-  and  £>=-^j 

7  49      116,     72     , 

Hence,  x=^+^(^^+—)=-=^. 

Or  comparing  the  given  equation  with  the  2d  form,  (jB.) 
a=9,  6=7,  c=116. 


74-s/ 49+4176      7+65      , 
Hence,  a:= jg ==-l8-=^- 


18.    Given    ^/10  +  a?— -Vl0+x=2,   to  find  the  value 
of  X, 


Considering  VlO  +  ^  ^s  the  quantity  sought,  and  com- 
paring the  equation  with  the  2d  form,  (^,)  we  have 
2^=1,6=2. 
1 


...VlO  +  a^=^+ V2-=2. 
10  +  0?= 2*=  16,  and  therefore,  x=6. 


Or  VlO+a?— 2=VlO+a7. 


Hence  by  involution,  10+a? — 4-v/10+j?+4=v/l0+ir. 


Hence,  14  +  07=  5  V 10+ a:. 
By  involution,  196 + 28a? +  0?^= 250 +  250?. 
Hence,  a?2+3a?=54. 
Comparing  this  with  the  1st  form,  (^,) 
2a=3,  and  Z>=54. 


19.  Given  a?+  \/5a?+ 10=8,  to  determine  the  value  of  a?. 


92  QUADRATIC  EQUATIONS. 

Multiplying  by  5,  and  adding  10  to  each  member, 

5a? -1-10  + 5  v/ 5^+10=:  50. 

Considering  \/5a?+10  as  the  quantity  sought;  and  com- 
paring the  equation  with  the  first  form,  (*/?,)  we  have 

2«=5,  and  6:^=50. 

. 5 

Hence  V^oo-\-lO^—^+^(60\^)=z5. 

By  involution  and  transposition,  5x=26 — 10. 

Therefore,  x=3. 
Otherwise. — By  transposition,  ^/5J?+10  =  8 — x. 
By  involutio-n,  5a?4-10=64' — 16x+x'^j 

Or  ^^—2107=— 54. 
Comparing  this  with  the  3d  form,  (j5,) 
a=l,  Z>=:21,  and  c=54. 

-_                 21dbv/441— 216     _        ^^ 
Hence,  07= =18  or  3.* 

20.  Given  >     ^        %«  ^    ^   j  j 

^  x  +  y=12j  to  find  07  and  y. 

Comparing  the  1st  equation  with  form  1,  (.^.) 

a=i,  b=12Q0. 

Hence,  07^=— -i  +  \/(^  +  1260)=:35. 

Tf  in  this  equation,  We  substitute  for  y,  its  value,  12 — x, 
found  from  the  2nd  we  have 

1207—07^=35  or  07^— 12o7=— 35. 
Comparing  this  with  third  form  (j1) 

a=6,  and  6=35. 
Hence,  o?=6dbl  =  7  or  5. 

One  of  which  being  taken  for  o?,  the  other  will  be  the 
value  of  y, 

*  The  ambiguity  is  introdnced  into  this  solution  by  the  involution 
of  the  given  equation;  for  (8 — xY=^{p^ — 8)^>*  and  one  of  the  values 
of  a;  corresponds  to  the  supposition,  that  52;-f-lO=(a; — 8)^,  and  the 
other  to  the  true  one,  that  52;+ 10=  (8 — rr)^. 


QUADRATIC  EQUATIONS.  93 

21.  Given  }  'Z2      ~Z' V  to  find  the  values  of  x  and  y. 

t     xy=S        ) 

Comparing  the  1st  equation  with  form  2d,  (J?,) 

a=3,  h=b,  c=2. 

„         X    5+V/25+24     ^ 

Hence,  -= -7; = 2. 

y  6 

.-.  x=2y,  and  2^2=8. 

o 

...  y—^-z=^1^  and  a?=4'. 

22.  Given  \  ^'+^0^7  ■^?'  ^  to  find  x  and  y. 
Assume  vy=x,  then  v-y^=x^,  and  vy^=xy, 

.•.  'z;2y'3_|_^2/^=12,  or2/2=:-— — 

^^2__23/2=l,  ory2=^j— ^ 

12  1 

Hence,— --—=  — ^,  or  122? — 24.=2?2^v. 

Therefore,  v^ — llv= — 24. 

Comparing  this  with  form'  3d,  (^,)  a=-^,5=24. 

Whence,  2;=^±v'(^— 24)=8  or  3. 

Otherwise. — Comparing  the  first  equation  with  12  times 
the  second,  x'^-{-xy=12xy — 2%^, 
.'.  x"^ — llxy=^'24^y^. 
Hence,  considering  3/  as  a  given  quantity. 

11±5 
By  form  3.  (B.)  x=—^y=Sy  or  3y. 


94 


QUADRATIC  EQUATIONS. 


Whence,  y=iy/6  or  1,  and  x=j^/6  or  3. 

23.  Given  }  y      x  >  to  find  co  and  y, 

(x+y—12,     ) 
Assume  x=vy, 

Then  v''y-\-^-=:28=a. 

vy-\-y=l2—b, 
a  b 

Multiplying  by  z^^-h  1. 


By  form  3.    (J5,)  v- 


5+«±V«^+2a6— 36^ 


26 


:3  or  §. 


Hence,  y=3  or  9. 
Wherefore  a:=9  or  3. 


24-.      Given  < 


x^  ^2 


Assume  a?+2/=5,  xy—p, 
-a — 5. 


Then  ^V^^ 

And  multiplying  by  xy=p,  x'^-{-y^=ap — sp. 
But,  x^+y^=s^^3sp.* 

^x-\-y=Sf  xy=p. 
From  the  square  of  the  first  equation  subtract  twice  the  second, 

"From  this  equation  subtract  the  second,  and 

x"^ — xy  -\-  jr2=:  ^3 — 2p. 

Multiply  by  the  first,  and  x^-\^y'=s^ — 3s/^. 


QUADRATIC  EQUATIONS.  95 

Again,  x'^-\-y^=zs^ — 2/?. 
Also,  ~Jr^^=(a-^sy—'lp  =  a^—^as^s^—^p. 

Hence,  — +  x'^  +  y^  ^^^=a^ — 2as-i-2s^ — 4<p=b. 

By  substitution,  (C,)  a^ — 2as  +  2s^ ?r=^- 

Clearing  the  equation  of  fractions,  «^- — 2as~=ab'{'2bs, 
b      a^ — b 


a  2 

— b  ¥      a?—b^ 

Consequently,    ^=0^  + v^(4^3  +  -2~) 

53 
Whence,    p-=^  becomes  known. 

Now,  {x — y )^  =  {x  +  yy — 4a?2/ = 5^ — 4p . 
Or  X — 3/==±=a/52 — 4p^ 


and  2/— 


2  ' ^  2 

Questions  producing  quadratic  equations, 

1.  Required  to  find  two  numbers,  such  that  their  sum 
added  to  their  product  shall  be  47  j  and  the  sum  of  the 
squares  shall  be  74. 

Let  X  and  y  denote  the  numbers. 

Then  a?  +  v  +  ^V=4^7,  ) ,      .,      -,  , 
A.r.Ax4}Lll,       '5  by  the  data. 

Add  twice  the  first  equation  to  the  second,  and 
a?24. 2a?y +  2/2^  2a?  + 23/=  168. 
Or,  (a7  +  2/)^+2.(a?  +  2/)=168. 
Hence  form  1,  (^.)  a;+j^=--l  +  13=12. 


96  QUADKATIC    EQUATIONS. 

This  equation  subtracted  from  the  first,  leaves 

By  subtracting  twice  the  last  equation  from  the  2d, 

X- — 2xy-\-y^=4f 

Hence  by  evolution,  x — y=d=:2. 

.•.07=7  or  5. 

2.  A.  and  B.  sold  130  ells  of  silk,  for  42  crowns ;  40  ells 
belonged  to  A.  and  90  to  B.  Now,  A.  sold  for  a  crown  i 
of  an  ell  more  than  B.  did.  How  many  ells  did  each  sell 
for  a  crown  1 

Let  X  =  the  number  B.  sold,  >  r 

Then  a:  +  ^  =  the  number  A.  sold,  \  ^"^^  ^  ^^^^^' 

90 
.*.  — =number  of  crowns  received  for  B.'s  silk. 

X 

*        n       40  120  ,  .,..,.„ 

And  — r-r=-^ 7= number  received  for  A.'s  silk. 

a?4-3     3a:+l 

90       120  _ 

•'•^"^3¥+l""^^' 

270a?+ 90  +  120a?=  126a?3  4-42a?. 

By  transposition  and  division,  21a:- — 58a?=15. 

Comparing  this  with  form  2,  (B^)  we  find 

a=21,  6=58,  c=15. 


V 1260  +  3364 -4-58     126     ^ 
Hence  a?  = — : ==.——=3,    number     of 

42  42        ' 

ells  B.  sold  for  a  crown. 

.'.  3  J,  number  A.  sold  for  a  crown. 

3.  There  are  two  square  yards,  paved  with  stone,  each 
stone  being  a  foot  square ;  the  side  of  one  yard  exceeds 
that  of  the  other  by  12  feet,  and  the  number  of  stones  in 
the  two  is  2120  ^  what  are  the  sides  1 

Ans.  26  and  38  feet. 


QUADRATIC  EQUATIONS.  97 

4.  A  laborer  dug  two  trenches,  one  of  which  was  6 
yards  longer  than  the  other,  for  £11  16s.,  and  the  digging 
of  each  cost  as  many  shillings  per  yard  as  there  were  yards 
in  its  length  ',  what  was  the  length  of  each '? 

Ans.  10  and  16  yards. 

5.  A.  and  B.  set  out  from  two  towns,  which  were  dis- 
tant from  each  other  247  miles,  and  traveled  till  they  met. 
A.  traveled  9  miles  a  day,  and  the  number  of  miles  walked 
by  B.  in  a  day,  increased  by  3,  was  equal  to  the  number  of 
days  occupied  by  the  journey.  Kequired  the  number  of 
days  they  were  traveling,  and  the  number  of  miles  passed 
over  by  each  1 

Ans.  13  days;  and  A.  went  117  miles,  and  B.  130. 

6.  A.  and  B.  having  bought  41  oxen,  for  which  each  of 
them  paid  420  dollars,  find  A.'s  worth  a  dollar  a  head  more 
than  B.'s;  how  must  they  divide  themi 

Ans.  A.  20,  B.  21. 

7.  Divide  the  number  14  into  two  parts,  whose  product 
shall  be  48.  Ans.  the  numbers  are  6  and  8. 

8.  Given  the  sum  of  two  numbers=9,  and  the  sum  of 
the  squares =45  5  what  are- the  numbers'? 

Ans.  6  and  3. 

9.  What  two  numbers  are  those,  whose  sum,  product, 
and  the  difference  of  their  squares,  are  equal  to  each  other  I 

Ans.  |  +  5\/5,  and  ^  + W5. 

10.  There  are  four  numbers  in  arithmetical  progression, 
of  which  the  product  of  the  two  extremes  is  22,  and  that 
of  the  means  40  ;  what  are  the  numbers  1 

Ans.  2,  5,  8,  and  11. 

11.  There  are  three  numbers  in  geometrical  progression, 
whose  sum  is  7,  and  the  sum  of  their  squares  21 ;  what  are 
the  numbers  1  Ans.  1,  2,  and  4. 

12.  Required  to  find  two  numbers,  such  that  the  less 
may  be  to  the  greater,  as  the  greater  is  to  12 ;  and  the  sum 
of  the  squares  may  be  45.  Ans.  3  and  6. 

13.  What  two  numbers  are  those,  whose  difference  is  2, 
and  the  difference  of  their  cubes  98  1  Ans.  3  and  5. 

9 


98  QUADRATIC   EQUATIONS. 

l^.  What  two  numbers  are  those,  whose  sum  is  6,  and 
the  sum  of  whose  cubes  is  72 1  Ans.  2  and  4. 

15.  Required  to  find  two  such  numbers,  that  their  pro- 
duct shall  be  20,  and  the  difference  of  their  cubes  61 1 

Ans.  4  and  5. 

16.  Required  to  divide  the  number  5  into  two  such 
parts,  that  if  each  part  be  divided  by  the  other,  the  sum  of 
the  quotient  shall  be  2J.  Ans.  3  and  2. 

17.  Divide  12  into  two  parts,  so  that  their  product  may 
be  equal  to  8  times  their  difference.  Ans.  8  and  4. 

18.  There  are  two  numbers,  the  sum  of  whose  squares  is 
89,  and  their  sum,  multiplied  by  the  greater,  is  104  j  what 
are  the  numbers  % 

Ans.  \^s/2,  and  -|V2,  or  8  and  5. 

19.  What  number  is  that,  which  being  divided  by  the 
product  of  its  two  digits,  the  quotient  is  5^  ;  but  when  9  is 
subtracted  from  it,  the  remainder  is  expressed  by  the  same 
digits  in  an  inverted  order  1  Ans.  32. 

20.  Required  to  divide  13  into  three  such  parts,  that 
their  squares  may  be  equi-different,  and  the  sum  of  those 
squares  may  be  75.  Ans.  1,  5,  7. 

21.  The  sum  of  three  equi-different  numbers  is  12,  and 
the  sum  of  their  4th  powers  962;  what  are  the  numbers "{ 

Ans.  3,  4,  and  5. 

22.  There  are  three  equi-different  numbers,  such  that 
the  square  of  the  least,  added  to  the  product  of  the  other 
two,  makes  28;  but  the  square  of  the  greatest,  added  to 
the  product  of  the  other  two,  makes  44;  what  are  the 
numbers  1  Ans.  2,  4  6. 

23.  Three  merchants,  A.  B.  C,  on  comparing  their 
gains,  finds  they  amount,  collectively,  to  1,444  dollars; 
that  B.'s  gain  added  to  the  square  root  of  A.'s  makes  920 
dollars;  and  that  B.'s  gain  added  to  the  square  root  of  C's 
makes  912  dollars;  how  much  did  they  severally  gain  1 

Ans.  A.  400,  B.  900,  C.  144. 


QUADRATIC  EQUATIONS.  99 

24.  What  two  numbers  are  those,  whose  sum,  added  to 
the  square  of  the  sum,  makes  702,  and  whose  difference, 
subtracted  from  the  square  of  the  difference,  leaves  56  1 

Ans.  17  and  9. 

25.  The  sum  of  two  numbers  is  10,  and  the  sum  of 
,  their  4th  powers  1552;  what  are  the  numbers  1* 

Ans.  6  and  4. 

26.  There  are  four  numbers  in  arithmetical  progression, 
the  common  difference  of  which  is  4,  and  their  continued 
product  9945  ,  what  are  the  numbers  1 

Ans.  5,  9,  13,  17. 

27.  The  sum  of  three  numbers  in  harmonical  proportion 
is  191,  and  the  product  of  the  first  and  last  is  4032,*  what 
are  the  numbers  1  Ans.  56,  63,  72. 

28.  The  sum  of  two  numbers  added  to  their  product 
makes  31;  but  the  sum  subtracted  from  the  sum  of  the 
squares,  leaves  48.     Quere  the  numbers  1 

Ans.  7  and  3. 

29.  Required  to  find  three  numbers  m  continued  pro- 
portion, whose  sum  shall  be  26,  and  the  sum  of  their 
squares  3641  Ans.  2,  6,  and  18. 

30.  Required  to  find  two  numbers  whose  product  shall 
be  320,  and  the  difference  of  their  cubes  to  the  cube  of 
their  difference  as  61  to  1 '?  Ans.  20  and  16. 

31.  If  700  dollars  be  divided  among  four  persons,  so 
that  their  shares  may  be  in  geometrical  progression,  and 
the  difference  of  the  extremes  to  the  difference  of  the 
means  as  37  to  12,  what  will  be  the  several  shares  1 

Ans.  108,  144,  192,  and  256  dollars. 

*  In  examples  of  this  nature,  we  may  assume,  letters  to  denote  the 
half  sum  and  half  difference  of  the  required  numbers ;  whence,  expres- 
sions for  the  numbers  themselves  are  readily  obtained,  and  these  being 
involved,  and  the  powers  added  together,  the  odd  powers  of  the  un- 
known quantity  will  disappear.  Hence,  if  the  unknown  quantity  does 
not  rise  higher  than  the  5th  power,  the  solution  may  be  effected  by 
qnadratic  equations. 


100  QUADRATIC  EQUATIONS. 

32.  The  sum  of  two  numbers  is  11,  and  the  sum  of 
their  5th  powers  17831 5  quere  the  numbers  1 

Ans.  7  and  4. 

33.  The  difference  of  two  numbers  is  8,  and  the  differ- 
ence of  their  4th  powers  is  14560  5  required  the  numbers  1^ 

Ans.  11  and  3. 

34.  What  number  is  that,  from  the  square  of  which,  if 
9  be  subtracted,  and  the  remainder  be  multiplied  by  the 
number  itself,  the  product  will  be  80 1  Ans.  5.    . 

35.  Required  a  number,  from  the  square  of  which,  30 
being  deducted,  and  the  remainder  multiplied  by  the  nmn- 
ber  itself,  the  product  shall  be  56  1  Ans.  2+3  x/ 2. 

36.  The  continued  product  of  five  equi-different  num- 
bers is  945,  and  their  sum  25 ,  what  are  the  numbers'? 

Ans.  1,  3,  5,  7,  and  9. 


*In  this  example,  assuming  for  the  half  sum  and  half  difference, 
we  obtain  a  cubic  equation,  whence  it  appears  that  questions  of  this 
nature  are  not  generally  solvable  by  quadratics.  When,  however, 
we  have  an  equation  of  the  form  x^-\-ace=b ;  if  b=mri,  and^^-j- 
ff=n,  the  equation  may  be  reduced  to  a  quadratic.  For  multiplying 
by  a:^,  and  adding  m'^^^  to  each  member,  the  given  equation  becomes 
cc^-\'ax^^m2x^=m^x'2-\-bXj  or  cc^'^-nx^=m'^x'2'\-?imx,    whence    by 

completing  the  squares  and  extracting  the  roots,  x^-^-='mx  ^~  ,  or 


In  the  above  example,  x  being  made  =  half  the  sum  of  the  re- 
quired numbers,  we  find  x^-j;-l6x=4S5=l  X  ^^i  where  7^+16= 
65,  whence,  x=l.  If  x^ — ax=mn,  and  m^ — a=7i ;  or  if  .-^^ — ax 
= — m?f.,  and  a — m^=7i,  the  equation  may  be  reduced  to  a  quadratic, 
and  X  found  =m,  as  before.  But  if  x^ — ax=miif  and  a — m^=n,  or 
x^ — ax= — 7n?tf  and  m^ — a=n;  the  equation  may  be  changed  to  a 
quadratic,  from  which  a  second  quadratic  x^ — mx=z^n,  will  arise, 

whence  x= : — . 


j.e^ 


v^ 


PROMISCUOUS    EXAMPLES.  101 


37.  The  sum  of  three  numbers  in  geometrical  progression 
is  35,  and  the  mean  is  to  the  difference  of  the  extremes  as 
2  to  3  ;  what  are  the  numbers  \         Ans.  5,  10,  and  20. 

38.  The  sum  of  three  numbers  in  geometrical  progres- 
sion is  13,  and  the  product  of  the  mean  by  the  sum  of  the 
extremes  is  30  5  required  the  numbers  %    Result,  1,  3,  9. 

39.  There  is  a  number  consisting  of  three  digits  in  geo- 
metrical progression ;  the  number  itself  is  to  the  sum  of 
its  digits  as  124  to  7 ;  and  if  594  be  added  to  it,  the  order 
of  the  digits  will  be  inverted  5  what  is  the  number!* 

Ans.  248. 

Promiscuous  examples  to  exercise  the  foregoing  rules, 

1.  What  number  is  that,  to  the  double  of  which,  if  44 
be  added,  the  sum  will  be  equal  to  4  times  the  number 
proposed!  Ans.  22. 

2.  A  gentleman,  meeting  4  poor  persons,  divided  a  dol- 
lar among  them  in  such  manner  that  their  several  shares 
composed  an  equi-different  series ;  and  the^sum  given  to 
the  last  was  4  times  that  given  to  the  first.  Quere  their 
several  shares  %  Ans.  10,  20,  30,  and''40  cents. 

3.  A  sum  of  money  being  divided  among  6  poor  per- 
sons, the  second  received  10c/.,  the  third   14c?.,  the  fourth 

*  Adfected  quadratic  equations  maybe  reduced  to  simple  quadratics, 
and  solved  without  completing  the  square,  in  the  following  manner : 

If  the  highest  power  of  the  unknown  quantity  has  a  co-efficient, 
divide  the  whole  equation  by  it ;  then  assume  the  unknown  quantity, 
equal  to  another  unknown  quautity  with  half  the  co-efficient  united  by 
the  opposite  sign.  Substitute  this  new  value,  and  a  simple  quadratic  will 
arise.  Thus,  if  x^J^2ax=h,  assume  z — a=x,  then  z^ — 2az  +  a^=x'2, 
2az — 2a^=2ax,  Yience  z"^ — a^=x'^--\-2axz=b,  and  z=\/  bJ^a^y  hence 

x=zz — a= — a-\-  \/  b  J^a'^. 

In  like  manner,  the  second  term  of  an  equation  of  a  higher  order 
may  be  taken  away  by  assuming  a  new  unknown  quantity  with  |,  f , 
etc.,  of  the  co-efficient  of  the  next  term  united  by  the  opposite  sign, 
in  place  of  the  quantity  sought. 


102  PROMTSCUOUS   EXAMPLES. 

25c?.,  the  fifth  28c?.,  and  the  sixth  33d.  less  than  the  first. 
The  whole  sum  divided  was  10c?.  more  than  three  times 
what  the  first  received.  How  much  did  they  severally  re- 
ceive 1  Ans.  40c/.,  30c?.,  266/.,  15c/.,  12c/.,  and  Id. 

4.  A  mercer,  having  cut  19  yards  from  each  of  3  equal 
pieces  of  silk,  and  17  yards  from  another  of  the  same 
length,  finds  the  four  remnants  together  measure  142 
yards  3  what  was  the  original  length  of  each  piece  1 

Ans.  54. 

5.  A  grazier,  having  two  flocks  of  sheep,  containing  the 
same  number,  sells  39  from  one,  and  93  from  the  other, 
and  then  finds  one  flock  twice  as  numerous  as  the  other  5 
what  number  did  each  of  them  contain  at  first '? 

Ans.  147. 

6.  From  each  of  16  coins  an  artist  filed  the  value  off  20 
cents,  when  the  coins,  being  examined,  were  found  worth 
only  11  dollars  68  cents  5  what  was  the  original  value  of 
each]  Ans.  93  cents. 

7.  The  hold  of  a  ship,  containing  442  gallons,  is  emptied 
in  12  minutes  by  two  buckets ;  the  greater  of  which,  hold- 
ing twice  as  much  as  the  less,  is  emptied  twice  in  three 
minutes,  and  the  less  is  emptied  three  times  in  two  minutes. 
Quere  the  number  of  gallons  held  by  each  bucket  1 

Ans.  26  and  13. 

8.  A  trader  maintained  himself  for  three  years  at  an 
annual  expense  of  ^50,  and  in  each  of  those  years  aug- 
mented that  part  of  his  stock  which  was  not  expended  by 
i  thereof.  At  the  end  of  the  third  year,  his  original  stock 
was  doubled ;  what  was  that  stock  %  Ans.  ^740. 

9.  A  gentleman  having  a  rectangular  yard,  100  feet  by 
80,  purposes  to  make  a  gravel  walk  of  equal  width  lialf 
round  it,  so  as  to  occupy  one  fourth  of  the  ground.  What 
must  be  the  width  of  the  walkl  Ans.  11.8975  feet. 

10.  Eequired  to  find  a  fraction,  to  the  numerator  of 
which,  if  4  be  added,  the  value  will  be  h. ;  but  if  7  be 
added  to  the  denominator,  the  value  will  be  J  1 

Ans.  j%. 


PROMISCUOUS    EXANPLES.  103 

11.  What  two  numbers  are  those,  whose  difference,  sum, 
and  product  are  as  the  numbers  2,  3,  and  5  respectively  % 

Ans.  10  and  2. 

12.  A  vintner,  having  mixed  a  quantity  of  brandy  and 
water,  finds  that  if  he  had  mixed  6  gallons  more  of  each, 
he  would  have  bad  7  gallons  of  brandy  for  every  6  of 
water  5  but  if  he  had  mixed  6  gallons  less  of  each,  he 
would  have  had  6  gallons  of  brandy  for  every  5  of  water. 
Quere  the  number  of  gallons  of  each  1 

Ans.  78  of  brandy,  and  66  of  water. 

13.  A.  and  B.  together,  are  able  to  perform  a  piece  of 
work  in  15  days;  after  working  jointly  6  days,  B.  finishes 
it  alone  in  30  days ;  in  what  time  would  each  of  them 
singly  effect  itl         Ans.  A.  in  21|  days,  B.  in  50  days 

14.  A  company  of  smugglers  found  a  cave  which  would 
exactly  hold  their  cargo,  viz:  13  bales  of  cotton,  and  33 
casks  of  rum ;  but  while  they  were  unloading,  a  revenue 
cutter  appeared,  on  which  they  sailed  away  with  9  casks 
and  5  bales,  having  filled  f  of  the  cave ;  how  many  bales, 
or  how  many  casks  would  the  cave  contain  1 

Ans.  24  bales,  or  72  casks. 

15.  There  are  two  numbers,  whose  sum  is  to  their  dif- 
ference as  8  to  1,  and  the  difference  of  whose  squares  is  128  5 
what  are  the  numbers  1  Ans.  18  and  14. 

16.  Eequired  those  two  numbers,  whose  sum  is  to  the 
less  as  5  to  2 ;  and  whose  difference  multiplied  by  the  dif- 
ference of  their  squares  is  1351  Ans.  9  and  6. 

17.  A  merchant  laid  out  a  certain  sum  upon  a  specula- 
tion, and  found,  at  the  end  of  a  year,  that  he  had  gained 
£69.  This  he  added  to  his  stock,  and  at  the  end  of  the 
second  year  found  he  had  gained  as  much  per  cent,  as  in 
the  first.  Continuing  in  this  manner,  and  each  year  add- 
ing to  his  stock  the  gain  of  the  preceding,  he  found,  at  the 
end  of  the  fourth  year,  that  his  stock  was  to  the  sum  first 
laid  out  as  81  to  16.     Quere  the  sum  first  invested  1 

Ans.  ^138. 


I04f  PROMISCUOUS    EXAMPLES. 

18.  There  are  two  numbers,  whose  sum  is  to  the  greater 
as  40  is  to  the  less,  and  whose  sum  is  to  the  less  as  90  to 
the  greater 5  what  are  the  numbers'?       Ans.  36  and  24. 

19.  The  area  of  a  rectangular  parallelogram  is  960 
yards,  and  the  length  exceeds  the  breadth  by  16  yards ; 
what  are  the  sides  1  Ans.  40  and  24. 

20.  The  area  of  a  rectangular  parallelogram  is  480,  and 
the  sum  of  the  length  and  breadth  52.     Quere  the  sides  ] 

Ans.  40  and  12. 

21.  The  sides  of  a  right  angled  triangle  form  an  equi- 
different  series,  whose  common  difference  is  3 ;  what  are 
the  sides  1  Ans.  15,  12,  and  9. 

22.  There  are  three  numbers,  the  difference  of  whose 
differences  is  5,  their  sum  is  20,  and  their  continued  pro- 
duct 130  5  required  the  numbers!        Ans.  2,  5,  and  13. 

23.  The  sum  of  three  numbers  is  21,  the  sum  of  the 
squares  of  the  greatest  and  least  is  137,  and  the  difference 
of  the  differences  is  3.     Quere  the  numbers '? 

Ans.  4,  6,  and  11. 

24.  There  is  a  number  consisting  of  two  digits,  which, 
divided  by  the  sum  of  its  digits,  has  a  quotient  greater  by 
2  than  the  first  digit.  But,  the  digits  being  inverted,  and 
divided  by  the  sum  of  the  digits  increased  by  unity,  the 
quotient  is  equal  to  the  first  digit  increased  by  4.  Quere 
the  number  1  Ans.  24. 

25.  Eequired  to  find  four  numbers  in  geometrical  pro- 
gression, whose  sum  shall  .be  15,  and  the  sum  of  the 
squares  85 1*  Ans.  1,  2,  4,  and  8. 


a;2  y'2 

*  Note* — If  we  assume  x,y  to  denote  the  means,  — .  and  —  will  re- 

y  X 

present  the  extremes ;  the  problem  will  then  be  the  same  as  example 
21,  page  94.  The  following  solution  includes  an  expedient  which  may 
be  sometimes  used  with  advantage. 

Let  X  denote  the  first  term,  and  y  the  common  multiplier. 

Then  x-Yxy-i^xy'^-\-xy^=,l6=a;  x^-^-x^-y^  -\-  x'^y^ -{•x'^y^=S5=^b. 


PROMISCUOUS   EXAMPLES.  105 

26.  There  are  five  numbers  in  geometrical  progression, 
whose  sum  is  242,  and  the  sum  of  their  squares  29524  5 
what  are  the  numbers!  Ans.  2,  6,  18,  54,  162. 

27.  There  are  six  numbers  in  geometrical  progression, 
the  sum  of  the  extremes  is  99,  and  the  sum  of  the  other 
four  terms  is  90.     Quere  the  numbers  % 

Ans.  3,  6,  12,  24,  48,  and  96. 


o2 

Then  2:2=- 


But  from  the  nature  of  progressionals, 

y^  +  y^+7/+l=^-^',   and  2/6  +  3'4  +  S''  +  l=^^; 
The  above  eqjiation  may  therefore  be  expressed  thus, 

.^^(y— i)^^(y+i) 

And  dividing  the  numerator  and  denominator  of  the  first  member 
by  (y — i)j  and  clearing  the  equations  of  fractions, 

.•.a2(y4+l)==i(y4  4. 22/3  +  2^24-2^  +  1.) 
.-.  (a2— i)y4— 2iy3__25y2_25y  -f-  (a2— 3)=0. 

...  (a2_5)y2-«23y— 25— 2i.-  +  (aS— 3)— =0. 

y  y^ 

Now  (3/4-^)2=3^24.24-^3 

.•.(a2— J)(y4-_)3;=(a2— 5)y2    f    2(a2— 5)   +  (o^— 5)— 

y  y^ 

Hence,  {a^—1)){yJ^—Y—2h{y-\'-)=:2a%  a  quadratic  equation. 

y  y 

1  1  .1      174-53     5 

22/2— 5y=— 2,    3/=^li-— -=2ori. 

Hence  1,  2,  4,  8,  or  8,  4,  2,  1>  are  the  numbers  sought. 

The  26th  and  27th  examples  may  be  solved  in  a  similar  way- 


106  RATIOS. 

28.  The  American  dollar  consists  of  1485  parts  by 
weight  of  pure  silver,  and  179  of  copper.  Quere  the 
specific  gravity  of  this  dollar,  the  specific  gravity  of  pure 
silver  being  11092,  and  that  of  copper  9000] 

Ans.  10821. 

29.  A  person  having  set  out  from  a  certain  place,  travels 
one  mile  the  first  day,  two  the  second,  etc.,  in  arithmetical 
progression.  In  six  days  another  sets  out  from  the  same 
place,  and  travels  in  the  same  direction  at  the  rate  of  15 
miles  a  day.     In  how  many  days  will  they  come  together  1 

Ans.  In  9,  and  in  20  days  after  the  first  sets  out. 

30.  Heavy  bodies  near  the  earth's  surface  are  known  to 
fall  16yi^  feet  in  the  first  second  of  time ;  and  to  pass  over 
S'paces,  which,  reckoned  from  the  commencement  of  the  fall, 
are  as  the  squares  of  the  time.  Now,  a  haiJU  stone,  being 
observed  to  descend  595  feet  during  the  last  Second ;  quere 
how  long  was  it  falling,  and  from  what  height  did  it  de- 
scend] Ans.  19  seconds,  and  5806  feet. 

31.  The  arithmetical  mean  of  two  numbers  exceeds  the 
geometrical  mean  by  13 ;  and  the  geometrical  mean  ex- 
ceeds the  harmonical  mean  by  .12.  What  are  the  num- 
bers 1  Ans.  234,  and  104. 

32.  The  fore  wheel  of  a  carriage  makes  six  revolutions 
more  than  the  hind  wheel  in  going  120  yards ;  but  if  the 
circumference  of  each  wheel  be  increased  one  yard,  it  will 
make  only  four  revolutions  more  than  the  hind  wheel  in 
going  the  same  distance.  Eequired  the  circumference  of 
each  ]  Ans.  The  fore  wheel  4  yards,  the  hind  5. 


Section  VIU. 
ON  EATIOS. 

68.  Ratio  is  the  relation  that  two  quantities  of  the  same 
kind  bear  to  each  other. 


RATIOS.  107 

Ratio  is  estiiilated  by  the  quotient  arising  from  the  divi- 
sion of  the  first  term  by  the  second.     Thus,  if  7=-,  ^  a  is 

said  to  have  the  same  ratio  to  h  that  c  has  to  d;  or  the 
quantities  a,  b^  c,  c?,  are  termed  proportionals ;  which  is 
briefly  expressed  thus,  a  :  b  : :  c  :  d. 

The  first  and  third  terms  are  called  antecedents,  the 
second  and  fourth  consequents,^ 

69.  When  four  quantities  are  proportionals,  the  product 
of  the  first  and  fourth  is  equal  to  the  product  of  the  second 
and  third ;  and  reciprocally. 

a     c 
If-=-,  by  multiplying  by  bd,  ad=bc, 

^,      .J,     y     T     cid     be  a     c 

Eeciprocally,  if  aa= be,  -r-j=-r-j,  or  'j^-j* 

70.  When  four  quantities  are  proportionals,  the  sum  of 
the  first  and  second  is  to  the  second,  as  the  sum  of  the  third 
and  fourth  is  to  the  fourth. 

^^a     c     a      ^      c      _^        a+b     c  +  d 

Or,  a+b  :  b  : :  c+d :  d. 

If  the  word  difference  be  substituted  for  sum,  the  propo- 
sition will  still  be  a  true  one. 

a     c  a  c  a^b     c^s^d 

Or,  a^b  :  b  :  :  C'^d :  d, 

71.  If  four  quantities  are  proportionals,  the  sum  of  the 
first  and  second  is  to  their  difference  as  the  sum  of  the  third 
and  fourth  is  to  their  difference. 


*  The  terms  are  similarly  designated  when  more  then  four  are  con- 
cerned. 


108  RATIOS. 

For  by  the  last  article,  —j — =—7—,   and  -^=—7—, 

And  dividing  the  former  by  the  latter, 

a  +  b     c  +  d 

7= ,.  or  a 4-0  :  a^o  : :  cA-d:  c^d, 

a^b      C'^d 

72.  When  any  number  of  quantities  are  proportionals, 
as  one  antecedent  is  to  its  consequent,  so  is  the  sum  of  all 
the  antecedents  to  the  sum  of  all  the  consequents. 

Let  a:b  ::  c  :  d: :  e:f: :  g  :  h,  etc.,  then  (art.  69.) 

ad=bc,  af=be,  ah=^bg^  etc.,  also  ab=^ba. 
.*.  ah-\-ad-\-af+ah^  etc.     =zba-\-bc-\'be-\-bg^  etc. 
Or,  aX  fZ) +  (/+/+ A,  etc.}  =6x  Ja-f-c+e+g*,  etc.} 
.•.(art.  69,)  a  :  6  ::  a+c  +  c+g,  etc.    :  6  +  c?+/+A,  etc. 

73.  When  four  quantities  are  proportionals,  if  the  first 
and  second  be  multiplied  or  divided  by  any  quantity,  and 
likewise  the  third  and  fourth,  the  resulting  quantities  will 
be  proportionals. 

^  ci     c  a     ma 

If  a  :  6  :  :  c  :  a, -r  =  -7,    but  7= — -,: 

^  b     d^  b     mV 

^     .c     nc         ma    nc  '        .  . 

And  -,= — ,,  .'.  —7= — "75  or  ma  :  mb  :  :nc:  nd, 
d     nd         mb     nd 

The  demonstration  is  manifestly  applicable  when  m  and 
n  are,  one  or  both,  fractional  numbers. 

74.  When  the  first  and  third  of  four  proportionals  are 
multiplied  or  divided  by  any  number,  and  also  the  second 
and  fourth,  the  resulting  quantities  are  proportionals. 

a     c         ma     mc         ma     mc 
b~~ d     '  '  b       d  nb     nd"* 

Or  ma  :nb  ::mc  ind,  m  and  n  being  any  numbers,  either 
integral  or  fractional. 

75.  If  four  quantities  are  proportionals,  the  like  powers 
or  roots  of  these  quantities  will  be  proportionals. 


If  a:b  ::  c:  d.  then  i  =  -7,  andr- = 
'  h     d^        0"" 


RATIOS.  109 

a     c  a""      e 


1 


Also,  —=— 5  or  a" :  6" : :  c" :  c?^, 
6^"     d- 


1      I       X      1 

And  aF:b'':i6^:d'', 

76.  A  ratio  compounded  of  several  ratios  is  indicated  by 
the  continued  product  of  the  quantities  which  denote  the 
component  ratios.  Thus,  the  ratio  compounded  of  the 
ratios  of  a  :  6,  of  c  :  c?,  and  of  e  : /*,  is  indicated  by 

ace  ace 

y^T^/'''^  Vdf 

The  ratio  which  the  first  of  a  series  of  quantities,  of  like 
kind,  has  to  the  last,  is  the  same  as  the  ratio  compounded 
of  the  ratios  of  the  first  to  the  second,  of  the  second  to  the 
third,  etc.  to  the  last. 

Let  «,  6,  c,  c?,  e,/*,  be  quantities  of  a  like  kind,  then 
a       a       b        c       d       e 
f       b       c       d      e      f 

77.  In  tw^o  ranks  of  proportionals,  if  the  corresponding 
terms  be  multiplied  together,  the  products  will  be  propor- 
tionals. 

Let  a:b::  c:  d^  and  e  :f: :  g  :  h, 

^^^^'    b~  d'   f~h      '''bfdh' 

Or,  ae,  :bf::cg:  dh. 

The  demonstration  may  be  easily  applied  to  any  number 
of  proportions. 

78.  A  ratio  compounded  of  two,  three,  four,  etc.,  equal 
ratios,  is  called  the  duplicate,  triplicate,  quadruplicate,  Qic, 
of  one  of  the  component  ratios. 

10 


1 10  RATIOS. 

The  ratio  compounded  of  any  number  of  equal  ratios  is 
the  same  as  the  ratio  of  such  power  of  the  first  term,  as  is 
indicated  by  the  number  of  component  ratios  to  a  like 
power  of  the  second. 

a       h 

Let  a  :  h  : :  h  \  c.  then  -7-= — 

^  b       c 

a^     ah      a 

0^    be      c  '      ' 

Again,  let  a  :  b  : :  b  :  c : :  c  :  d : :  d :  e,  etc. 

aaabacadae 
Then,  y=y,  y=y»  T"^i:'  T^^T'  y=-yr,etc. 

to  7^,  equations  5  and  multiplying  the  first  and  second  mem- 
bers respectively  together, 

a^      a       b       c       d       e       a 

Or,  a'':b''::a :/. 

79.  When  the  ratio  of  the  first  of  three  quantities  to  the 
second  is  the  same  as  the  ratio  of  the  second  to  the  third, 
the  ratio  of  the  first  to  the  second  is  termed  the  sub-dupli- 
cate of  the  ratio  of  the  first  to  the  third. 

Whec^four  quantities  are  continued  proportionals,  the 
ratio  of  the  first  to  the  second  is  called  the  sub-triplicate 
of  the  ratio  of  the  first  to  the  fourth. 

A  ratio,  compounded  of  a  simple  and  sub-duplicate  ratio, 
is  called  a  sesquiplicate  ratio. 

80.  The  sub-duplicate  ratio  is  equivalent  to  the  ratio  of 
the  square  roots ;  the  sub-triplicate,  to  the  ratio  of  the  cube 
roots ;  and  the  sesquiplicate,  to  the  ratio  of  the  square  roots 
of  the  cubes. 


Let  a  :  b  : :  b  :  c  : :  c  :  d.         Also,  b:e::  e:c, 

h 
a       (L  a      a 

First,  (art.  78,)  y=y ,  •-.  T^l  ^^^  ^ '  ^  •  *  ^ 


c^ 


VARIATIONS.  Ill 

i 

a 
b  """ 


In  like  manner, -^=-7,  ora:5: :  a"^    :  d 


1 


Also,  since  5  :  e  : :  e  :  c,  and  a  :  b  : :  c  :  d, 

e'^=bc=ad.  (art.  69.)  But  r— =-7-= — -7=-t-> 
'  ^  ^  b"^       d      ad     e^ 

3 

.•.  -— = — ,   or   a^  :b^  ::a:  e,  in  which  the  ratio  of 

a  :  e,  is  compounded  of  the  ratios  of  a:b,  and  of  6  :  e,  or  of 
the  ratio  of  a:b,  and  the  sub-duplicate  of  the  ratio  of  5  :  c. 


ON  THE  VARIATIONS  OF  QUANTITIES. 

81.  In  the  investigation  of  the  relation  which  varying 
and  dependent  quantities  bear  to  each  other,  the  conclu- 
sions are  more  readily  obtained,  by  expressing  only  two 
terms  in  each  proportion,  than  by  retaining  the  four.  But 
though,  in  considering  the  variations  of  such  quantities, 
two  terms  only  are  expressed,  it  must  be  remembered  that 
four  are  supposed ;  and  that  the  operations,  by  which  our 
conclusions  are  obtained,  are  in  reality  the  operations  of 
proportionals. 

82.  One  quantity  is  said  to  vary  directly,  as  another, 
when  one  is  such  a  function  *  of  the  other,  that,  if  the  for- 
mer be  changed,  the  latter  will  be  changed  in  the  same 
ratio.  Thus,  if  B  be  such  a  function  of  .5,  that  by  chang- 
ing Jl  io  a,  B  shall  be  changed  to  b  ;  making  A  i  a  i  :  B  :b^ 

=*  The ///7zc/io/j  of  any  variable  quantity  ic,  is  an  algebriac  expres- 
sion, in  which  x,  combined  with  invariable  quantities,  is  involved. 
Thus,  l  +  ic,  (l-fa;)^,  ax,  iP",  «^,  etc.  are  functions  of  x.  Analysts 
sometimes  use  the  Greek  letter  0,  to  denote  a  function.  Thus^  (^x  may 
represent  any  function  of  x. 


112  VARIATIONS. 

A  is  said  to  vary  directly  as  J5.     This  relation  is  designated 
thus,  Ji^B, 

83.  One  quantity  is  said  to  vary  inversely  as  another, 
when  the  latter  is  such  a  function  of  the  former,  that  the 
one  being  increased  or  diminished,  the  other  will  be  dimi- 
nished or  increased  in  the  same  ratio.  Thus,  if  B  be  such 
a  function  of  ^,  that,  by  changing  A  io  a^  B  becomes 
changed  to  h ;  making  A  \a::h  i  B,     Then  A  is  said  to 

vary  inversely  as  ^.     Indicated  thus,  Aac -^ , 

84*.  One  quantity  is  said  to  vary  as  two  others  jointly, 
when  the  two  last  are  such  functions  of  the  first,  that  the 
ratio,  which  any  two  values  of  the  first  bear  to  each  other, 
shall  be  the  same  as  the  ratio  compounded  of  the  ratios  of 
the  corresponding  values  of  the  other  two.  Thus,  A  varies 
as  B  and  C  jointly,  (^AccBCj)  when  A  being  changed  to 

ABC 

a,  B  chang-es  to  5,  and  C  to  c,  so  that  — =-7-X  — . 
°  a       0       c 

85.  One  quantity  is  said  to  vary  directly  as  a  second, 
and  inversely  as  a  third,  when  the  ratio  which  any  two 
values  of  the  first  bear  to  each  other,  is  the  same  as  the 
ratio  compounded  of  the  direct  ratio  of  the  corresponding 
values  of  the  second,  and  the  inverse  ratio  of  those  of  the 
third. 

Thus,  A  varies  directly  as  B,  and  inversely  as  C, 
(Aoc  YTf)  when  A,  B^  C ;  a,  Z>,  c,  being  corrresponding 

A     B       c 
values, — =-7-X77-. 
a       0      C 

In  the  following  articles.  A,  J5,  C,  etc.  represent  cor- 
responding values  of  any  quantities,  and  a,  6,  c,  etc.  any 
other  corresponding  values  of  quantities. 


VARIATIONS.  113 

86.  If  A^B^  and  m^  n  denote  any  given  numbers, 

-n 

Then  A^xmBo: — , 

^      .        A     B  B     mB     ^B     A     mB    ^B 

For  since —  =  i-,    and  -r-= — 7=7-"  5  — = — i:=^~r'* 
a       0  0       mo      ^if       a       mo      ^5 

87.  liAxB,  then  ^°x^%-  and  A^azB\     See  art.  75. 

88.  If  AazB,  and  CazD,  then  AC^BD.     See  art.  77. 

89.  \iA:f:BC,  then  -Soc^,  and  Coc-g-, 

^      A     B      C         Ac    B         B      A      c  ^^    A^ 

a       0       c  aC      0  ^        0       a      C  ^       C^ 

,,       Ab     C  C      A      b        ,^   A  ^^ 

Also, -^=— ,  or— =— X-H-,     (C'gc^.) 
^  aB      c'        c       a      B  '     ^      B  "^ 

From  the  three  preceding  articles  it  appears,  that  quan- 
tities connected  by  the  sign  oc,  may  be  treated  as  the  mem- 
bers of  an  equation,  as  far  as  multiplication,  division,  in- 
volution, or  evolution  is  concerned. 

90.  If  AccC,  and  J9xC,  A  and  B  being  quantities  of  the 
same  kind,  then  (^Ad:zB)QfCj  and  y/ABccC. 

^      .        A      C  B      C      A     B  A     a 

Forsince — = — ,  and  --,--= — :  — =-t-.    .*.  7r=Tj 
a       c'  b       cab  B      b 

^       A  a     ^       Ad^B     a±b         A±:B     B      C 

And-^d=l=T=bl,  or — ^— =— ^— .   .-. -— -^=— ==— 
B  b       ^  B  b  adub       b       c 

Again,  since  AccC,  and  B  ocC,  (art.  88,)  ABocC^, 

.-.(art.  87,)  s/AB^C. 

91.  If  while  A  and  B  vary,  AB=b,  constant  quantity, 

A  cx-g ,  and  B  oc-^ 
For  since  AB=ab^  — ~~»'    (•^^c">) 

10* 


114  SERIES. 

Sectiojt  IX. 
OF  SERIES. 

92.  From  the  nature  of  powers,  (art.  14,)  we  readily 
discover  that  any  two  powers  of  the  same  root,  multipli(:»d 
together,  produce  a  power  indicated  by  the  sum  of  the  ex- 
ponents of  the  factors ;   thus  a^.f^S—^S.   ^m^^n__^ni4-n. 

_      ,        ,      ,       ,   fl^^      la,a,a.a,a  aJ^ 

On  the  other  hand,  — =— ^ =  fla=a^--=ia"~'' 

,,       a"^       a"^        1 
Suppose  n==m  +  v,  then  —=-^=- 

But  in  this  case  m — n= — v^  if  then—  can  be  always  de- 
noted by  a"""";  a"""  must  be  equivalent  to  — 

Considering  the  consecutive  powers  of  a  given  root  as 
the  terms  of  a  geometrical  series,  extending  at  pleasure, 
above  and  below  unity. 

111. 

-  i  ~z->  "f  1?  ^5  ^^  ^^  etc. 

Or  the  equivalent  series,  a~^,  a-^,  a"^,  a^,  a\  a%  a^,  etc. 

The  index  denotes  the  order  of  the  term  beginning  with 
unity,  and  making  the  index  positive  or  negative,  accord- 
ing as  the  quantity  is  above  or  below  the  point  of  com- 
mencement. 

93.  Between  any  two  of  these  terms,  let  n — 1  mean  pro- 
portionals be  interposed,  ex.  gr.  between  a^  (=1?)  and  a, 
calling  the  first  x,  then  (art.  78,)  1 :  a: :  V  :  x"" : :  1 :  a?". 

Our  series  then  becomes,  (x'^  being  =  1  =a^,) 

'r~2Il  ^— n— 1        r^—U  rp—2       rn—1       /y)0        -yil        vt2 

...  a?",  a?°+S  ^""^^  ^'^"j  ^^"■^'S  etc. 


SERIES.  115 

Now,  the  product  of  any  two  terms  of  this  series  is  mani- 
festly indicated  by  x,  Avith  an  exponent  equal  to  the  sum 
of  the  exponents  of  the  factors,  as  a?°X  a^2''+^=a;5"+^5 

^.+1  X  0?-°=  -  -  -  X  —  — =:  0?^=  0?"+-^-", 

1       a?"      07° 

^2a+  3 

=:  a:'"+  i  =  ^2n-h3— (n-f-  2) 


If  for  X  and  its  powers  we  write  a^  and  its  powers,  the 
series,  though  changed  as  to  the  form  of  its  exponents, 
evidently  retains  the  same  essential  character,  and  the 
operations  of  multiplication  and  division  are  performed  by 
taking  the  sum  and  difference  of  the  exponents  of  the  fac- 
tors, whether  those  exponents  are  positive  or  negative,  in- 
tegral or  fractional.  By  thus  arranging  the  powers  as  the 
terms  of  a  geometrical  series,  we  perceive  that  the  expo- 
nent, whether  integral  or  fractional,  serves  not  only  to 
designate  the  power,  but  to  indicate  the  situation  of  the 
term,  in  relation  to  the  unit's  place. 

94.  From  these  principles  we  readily  infer,  that  any 
letter  or  quantity  may  be  removed  from  the  denominator 
of  a  fraction  to  its  numerator,  and  vice  versa,  by  changing 
the  sign  of  its  exponent. 

Thus,  ~^=—x—  =  ax-^. 
x^      1      x^ 

a-\-b  ,  ,  ,  ,        a 

^  ,  \„,,  ==(a  +  b).(a''—b^)-k.       ab''=  ,— 

95.  It  frequently  happens  in  the  division  or  evolution 
of  algebriac  quantities,  as  well  as  in  common  arithmetic, 
that  to  whatever  extent  the  process  may  be  continued,  a 
remainder  will  still  occur;  in  which  case  the  resulting 
quotient  or  root  mostly  assumes  the  form  of  an  infinite 
series. 


116  SERIES. 

Tlius^  ~ ■  —  l-\-x+x^-\-x^+x^-{-j  etc.  to  infinity. 

b^      h^        b^         b¥ 


^«^+A»=:«  +  2--8^3+i6^-T28^.+.  etc.* 

In  some  instances,  when  a  few  terms  of  the  series  are 
obtained,  the  law  of  continuation^  or  the  relation  of  the 
successive  terms  to  those  which  precede  them,  becomes 
manifest.  The  first  of  the  series  above  given  may  be 
readily  continued  to  any  proposed  extent.  The  law  of  con- 
tinuation in  the  last  is  not  obvious  on  first  view  5  it  will 
however,  be  shown  further  on. 

96.  In  the  investigations  connected  with  series,  the  me- 
thod of  indeterminate  co-efficients  is  often  found  particularly 
convenient.     It  depends  upon  the  following  theorem. 

Let  .^a7+5a?'2+Ca?^  +  Da?*,  etc.  =«a?4-^a?^  +  ca?^+^a?*,  etc., 
the  series  being  both  infinite,  or,  if  finite,  extending  to  the 
same  number  of  terms;  and  A^  B^  C,  a,  5,  c,  invariable; 
if  then  the  above  equation  be  true,  whatever  value  may  be 
assigned  to  a?,  A  will  be  equal  to  «,  B=b,  etc.  5  for  dividing 
by  a?,  we  have 

A  +  Bx+Cx'^+Dx^^  etc.=a'\-bx+cx'^-\-dx'^,  etc. 

Now,  the  equation  being  true  for  all  values  of  a?,  must 
hold  if  a?=0,  in  which  case  the  equation  becomes  A=a; 
subtracting  this  equation  from  the  given  one, 

Bx  +  Cx^'+Dx^,  eic.  =  bx-^cx^+dx'^,  etc. 
Dividing  by  x,  B-{-Cx-\-Dx\  eic,  =  b-{-cx+dx%  etc. 
Whence,  if  a?=0,  B  =  b, 

In  the  same  manner  C=c,  D=d^  etc.  m 

This  equation  becomes  by  transposition, 

Ax-\-Bx^'\'Cx''-\-I)x'-{',  etc.  )  _^. 
— ax — bx'^ — cx^ — c?a?*-i-)  etc.  )  "~ 

*  When    the  successive    terms  of  an   infinite   series  continually 
decrease,  it  is  called  a  converging  series ;  in  which  case,  the  sum  of  the    • 
series  may  be  approximated  by  collecting  a  finite  number  of  terms. 


SERIES.  /  117 

Hence  it  appears,  that  when  all  the  terms  of  a  general 
equation  are  brought  to  one  side,  the  co-efficients  of  the 
several  powers  of  the  unknown  quantity  are  respectively 
equal  to  0. 

This  principle  is  applied  in  the  following  examples. 

1.  Required  to  express  - — ^ :  in  a  series. 

^  ^         1 — 2a? +07^ 

It  is  easy  to  perceive  that  the  first  term  must  be  1. 

Assume  then  - — ^ — : — -=!-{■  ax ^bx^-\-cx^-\-dxK  etc. 
1 — 2x-\-x^  '         '       ' 

Multiply  by  1 — 2a? +a:^,  and  bring  the  terms  to  one  side 
of  the  equation 

Then,  l-\-ax-{'hx'^  -fca?^   +dx^  +,  etc.  1 

— 2a7 — 2aa?^ — 2Z)a?'^— 2ca7*— ,  etc.  V  =0 
— 1  +0:2      _pc5^3  _j_5^i  _p^  etc.  ) 

Hence,  «— 2=0,  h—1a-V  1  =  0,  c— 25-f  «=rO,  c?— .2c +  6=0. 
...  a=.2,  5=4—1  =  3,  c=6— 2=4,  c^=8— 3=5. 

Whence,  ^=l+2a?+3a?^+4a?'^+5a?^+,   etc.  in 

which  the  law  of  continuation  is  manifest. 


2.  Required  to  develope  ^/a^'{-x^^  in  a  series. 


V«^+a?^=V(«^Xl  +  -)=ax/(l+-)=aVl+2^% 

a?2 
Puttino;  -  =  2;2. 


Assume  -,/ l-\-z''—l+az'^-\-hz^+cz^-\'dz^-\-,  etc.* 

*  If  the  series  iJ^az-^-bz^  J^cz^^  etc.  had  been  assumed,  we  should 
have  found  a=0,  ^=0,  etc. 


1  ]  8  SERIES. 

By  involution  and  transposition, 

l-{.2az''+2bz^+2cz^'\-2dz^+2ez^''+,  etc.  ^ 

a''z^  +  2abz^+2acz^+2adz^^,  etc.  V  =  0. 
6^2:8 +26c2r^o,  etc.  3 
— 1 — z"". 

.-.  2a— 1  =0,  2b+a^=0,  2c+2ab=0,  2d+2ac  +  b'^=0,  etc. 

Whence,   «=2' ^^"S'^^Tg' ^^"128' '"^256 
11  1    „       5 


...  Vl+-^=l+5^--§-^+ je^^-j^8-^+256^^^  ^^^- 
Consequently, 

>•  /p3  'j^  oa8  ^^^  ^ 

Va^+x^  =  a  J   1  +  2^-8^.+  i6^B-li8^s'  etc.  j  = 

07^       o:*       0?^         So;'^      •  7a?^o 
'   ^+2^~8^3+i6^— 128^7+256^9'  ^*^- 

To  find,  if  possible,  the  law  of  continuation,  we  observe 
that 

1__1     —1     ^_1     —1     —3 

~B~2^T"'   16~2^ir^    6"' 

5       1     _i     -_3    _5       7       1—1     —3     —5 


X-J->^-7r-X-77-,      K^  —  nX-j-.X-^X- 


128""2      4         6^8'    256~2      4         6 

—7 
X  -T^»     Hence  the  law  is  evident 

97.  Eequired  to  express ^  in  a  series,  n  being  an  in- 
tegral number. 

0?— y      0?  1- -2r 


SERIES.  119 

y 

Assuming  2:=-. 
cc 

1 2^n 

Now,   if be  resolved  into  a  series,  it  is  manifest 

1 — z  ' 

the  first  term  will  be  1. 

And  therefore,  —  =  o?""^  X  -z resolved  into   a 

X — y  1 — z 

series,  will  have  its  first  term  a?°~^ 

a?" 

a:" — y"^     y^ — a?**     2/^  ^         _      1 — ^" 


Again,  • 


^—y     y—^    y       r  i— ^ ' 

1— - 
y 


making  v=-^  which,  resolved  into  a  series,  will  have  its 
first  term  y"~^. 

•vi^ y^ 

Whence  it  appears,  that  if be  resolved  into  a  se- 

X — y 

ries,  the  first  and  last  terms  will  be  x^~^^  and  y""-^  respec- 
tively. 

1 — z"" 
Assume  then    =  l+az-^-bz^ -{',.,, jp z""'^  -\-  qz^'-K 

Multiply  by  (1 — 2:,)  and  transpose  1 — z"",  whence, 

1+az+bz^  +  cz^j    pz^'-^+qz''-^  ^ 

— z — az- — bz^,     — mz""-^ — pz""-^ — qz%  >  =0 
—1  +z%  ) 

.-.    (art.   96,)   a— 1=0,   5—^=0,   c— 5=0,  p^m=:0, 
^— p=0,  1 — g=0.     Or  a=l,  b=l,  c=l,  etc. 

.-.  — -5-  =  a?^-'X  {l+z+z^-\-z\     2r^-3+2r°-'|  = 


120  SERIES. 

Whence  03^—^"=: (a? — y)X  [x^-^+x^'-^y+x^'-^y'^. 

Here  the  indices  of  x  descending  regularly  from  n — 1, 
to  0,  or  n — 7^,  it  appears  the  number  of  terms  in  the  series 
is  n, 

98.  Required  to  develope  (l+o?)''  in  a  series,  n  being  a 
whole  positive  number. 

It  is  easy  to  perceive  that  the  first  term  of  the  series  is 
1  ]  and  that  the  powers  of  x  regularly  ascend. 
Assume  then, 

(1  +  0?)"— l  +  aa?+Z)a?2+ca?'^  +  c/a?H-,  etc.  {A.) 
Consequently,  {l-\-yy=::l-\-ay-\-hy'^-\-cy'^-\-dy*^  etc. 
By  subtraction,  (l+a?)° — (l-^-yY—a^x — y)+h{x'^ — y^) 
+  c(x^ — y^)  4-  d(x^ — y%)  etc. 

Divide  by  1-f  a? — l+i/=a? — y,  and  we  shall  have 

X — y  ^  X — y  ^ 

e(^±=?(!)+<Z(^-=i:)  etc. 

^  X — y  '       ^  X — y  ^ 

Or  (l+a:)"-i4-(l+^r-^X(l+i/) (1+2/^"^ 

=ia-\-h{x-Yy)~\-c[x^~'-\-^y-\-y^) 
+(f(a?'^  +  a?'^2/  +  a?^2+?/3)+,  etc. 
This  equation  being  true,  whatever  value  may  be  as- 
signed to  X  and  y,  let  us  suppose  a:=y,  then  this  equation 
becomes 

(art.  97,)  n\\-\-xy-^z=za-\-'lhx-\-'^cx''-\-Ux^-\-bex',  etc. 
Multiply  by  (1  +  0?) 

.-.  71.(1  +  0?)°— a+2^a7+3ca;2+4ffir^  +  5eo?*,  etc. 
ao?+26o?2+3co?^  +  4c?o?*,  etc. 
But  (eq.  ^,)  multiplied  by  tz, 

71.(1 +o?)°=7z+wao?+7z5o?'2+7ico?3+wc?o?*,  ctc. 

_,  ,  ,        ^ — 1     ^-""1 

Consequently,  a=7z,  6)=«.— ^=7?.-^, 


SERIES.  121 

,  n — 2       n — 1  n — 2    ,      n — 3        n — 1  n — 2  n — 3 
n — 4        n — 1  n — 2  n — 3  n — 4 

Hence,  (a+J)''=a°X(l+-)''=: 

a«.Sl+«-+n.-^-+,etc.}= 

n — 1       ,         n — 1  n — 2 

aJ'-\-na^-'h-\-n,—^a''~^y^-^n,—^ -_a»-353  4-,  etc. 

When  n  is  B,  whole  positive  number,  as  here  supposed, 

the  series  is  finite ,  the  number  of  terms  being  n-\- 1 ;  for 

the  co-efficient  of  term  n-{-2is 

n — 1  n — 2     n — n 
n. — ^r—' — 5— ••••     ,  1=0. 

If  (a — by  be  required,  we  may  put  — b  and  its  powers 
for  +6  and  its  powers. 

n — 1 
Whence,  (a — Z>)°=a° — na''-^b+n—^a''"^b^ 

n — 1  n — 2       ,         n — 1  n — 2  n — 3 
^n.  —^ —a^-^¥-\-n,—^ g ^a"-*i*,etc. 

99.  Required  to  develope  (l+a?)°  in  a  series. 
Assume  (l+x)"=l+aa?+Ja7^+ca?3+(Za7*+ea?^,  etc.  (^.) 
Then  (l+^)^=l+«2^+^y'^+c?/34-c?2/+e?/5,  etc.  (J5.) 
Put  (l+a?)i='y,  (l+2/)r=^5  then  l+a?=t;°,  \^y^w'\ 
(l+a:)"=2;-,  (l+y)^=t^-. 

Substituting  these  expressions  in  equations  A  and  J?,  and 
subtracting,  we  have 

if^ — yf^ — aix^-y) + b(x''—y^)  +  c(a?*^— y^)  ^ 
(/.(a?* — 2/*)+,  etc. 


122  SERIES. 

.'.(art.  97,)  {v — w)X  [v'^-^+v'^-^w.    w'^-^l^ 
d^x-'+x^y  +  xy^+y^  etc.  |  (C.) 


But  X — y=l+x — 1+2/=:?;°— 2^°= (v — w)X 
.«.  Dividing  equation  C  by  this,  we  have 

c(a?2 + a7y  +  y^) + d(x^  +  a?^y  ■{■  xy^ + y^),  etc. 

Now,  suppose  x=y^  whence  v-^w  and  our  equation  be 
comes  (art.  97,) 


—  =(-.—)= a +  2507+ 3ca?2  +  46Za?3  + 560?*,  etc. 


712;°""^        '71   V 

Multiply  by  ?;°=l+07,  and  we  have 


711  W>  m 

— v'°=: — ,(l'i'Xy=a+2bx+3'cx^+4}dx^  +  6exS  etc. 

aa? + 2bx^  +  Sco?^  +  ^c^o?*,  etc. 
But  (equation  ^,) 

— .(l+a?)n= j ax  A hx'^A cx^-\ dxf^.  etc. 

Hence,  by  equating  the  homologous  co-efficients, 

m  m     ^ 

-1 


771  n  m  m — n  n 

n*  2         n     In  ^  \ 


"    -3 


m  m — n  m — 2n       ,        n  m  m — n  m — 2/i  m — 3n 

n      2n       3n     ^      '     *     4f  n     2n         3n        4^n 


SERIES.  123 


Hi  ta  O    m 

m  m  b      m  m — n  b^      m  m — n  m — 27i  b^ 

l(y).  In  the  preceding  Investigation  m  has  been  supposed 
to  be  aflSirmative ;  if  we  now  take  m  negative,  and  make 
the  same  assumptions  as  before,  we  have 

1        1      w'"" — v"^  1 

1 


V~^ W~^  = =  > = •  7)™ 7/»^  = 


(x—y).\a+b(x+y)  +  c.{x^+xy+y^)-{- 

d{x^  +  x"y  +  X2f'\'y^)  +  ,  etc.  |  = 

(o — w).  { ^;"~^  4-  v''"-w.     Tif-^  \ 

|a+6(a?+2/)  +  c(a?2  +  a?3^4-2/2)  +  ,  etc.  | 

Whence,  dividing  by  n,v — w^  and  assuming  x=y^ 

1       7n,v^~^ 
i^X  ='y'^-^|a  +  26a?+3ca?2-}-4c?a?3-{-,  etc.  I 

.*.  Multiplying  by  ?;,  and  putting  1-f-a?  for  v% 


m 


v-'^=z  1 4-  /rj  a4-2&a:+3ca:H4£fa73-f ,  etc.  |  = 

a'\-^'bx+^cx^  +  ^dx\  etc. 

aa?+26a?2  +  3ca:3,  etc. 

— m, 
Now,  multiplying  equation  A  by ,  and  equating  the 

homologous  co-efficients, 

— m         — m     — m — n 

7^    '  n  27i     ' 

— m    — m — n    — m — 2n    , 
c= X  — T. X  — 5 ,  etc. 


124*  SERIES. 

Whence  it  appears  that  the  formula  is  correct  whether 
the  index  is  positive  or  negative.  This  is  Newton's  cele- 
brated binomial  theorem. 

101.  By  the  aid  of  this  theorem,  a  binomial  maybe 
raised  to  any  power,  or  evolved  to  any  root  by  simple 
substitution. 

Several  examples  of  the  use  of  this  theorem,  when  the 
index  is  a  whole  positive  number,  are  given  in  section 
second. 

1.  Required  to  develope  V , sr^  in  a  series. 

^       a 
Here  m— — 2,  7i=3. 

.      x.-i        ,,      2     a?      2— 2— 3a:^     2—2—3 

.•.a(l  +  -)      =ajl  —  -. -5-. — ^ — .—  — ?r. — ^ — •. 

^       a^  <         3a3        6a^3D 

—2—6  x^     2  —2—3  —2—6  —2—9  ^         , 
2a?    5a?2    40a?'         llOa?^ 


2.  What  is  the  value  of  Va^+6  in  a  series  1 

b       ¥      563        105*      ^ 
Ans.  a+__— +— _^j3^-,etc 

3.  What  is  the  value  of  V— in  a  series. 

a^ — a?2 

1       a?^      3a?*      5a?9       35a?^      ^ 


SERIES.  125 

2 

4.  Required  the  value  of  (a — b)^  in  a  series. 
Result,  a^  ,\1 


5a     25a^     12oa^ 
26b*       4686^ 


625a*     15625^5'     ^ 


Ct  ■  I  /K 

5.  Required    V in  a  series.* 

Result,  1-] |-7r-a+2r^+5-7+5-Tj^'tc. 

6.  What  is(7 )^  in  a  series  1 

^b — X 

.c^.l  < .     3a?     12a?2     52a?3     234a?* 
Ans.y  .|l4.gjH-^,+^^3  +  g-^,  etc,} 


REVERSION  OF  SERIES. 

102.  Given  ^=a?+|  +  g+^-f^^,  etc.  to  in- 
finity,  to  find  x  in  terms  of  z. 


*  This  quantity  is  reducible  to 
11* 


126 


SERIES. 


Assume  x^az 4  hz'^  +  cz^ -\'dz^  + ,  etc. ' 
Then  -h=     -^-{-ahz^+acsi^^  etc. 

,  etc. 


2.3"" 

a?* 

2.3.4'' 

— ^r. 


2 
2:3 +T^*'  etc. 


2.3.4' 


=  0 


a=l,  5=—-^,  0=-^^—-^=-^,  c^=— -^  — -^  + 


8^4 


"2i^ 


4.- 


j^3        -j3        ^4 

Whence,  a?=2r— ^+^ — 7-+)  e*^*  where  the  law  of 


continuation  is  manifest. 

z^  z^ 


2.  Given  a?=2r- 
z  in  terms  of  x. 


2.3^2.3.4.5     2.3.4.5.5.7' 


;,  etc.  to  find 


Let  2r=aa7-f  5a?3+c^5+c^^ 
z^ 


Then  — . 


2.3" 


a3         0^6         ah^  ^ 

2:3A5'^'+2:3:4^'- 


a7 


2.3.4.5.6.7 


0' 


— X. 


=0 


SEEIES.  127 

__,   ,     J_    __1 1      _   1»3 

•'•  ^"'■^'  ^""2.3'^~2.3.2     2.3.4.5~2.4.5 

1  1.3         1  1  1.3.5 


^~^0  A.  Q'O  Q  F;        a.  Q  a.* 


2.4.9"^  2.8.5     4.9.4,  "^2.3.4.5.6.7     2.4.6.7 

cc^      1.3a:5     i.3.5a;7     i.3.5.7a?9 
Hence,  z=x+^^+^^^+^-^+^^^-^^,  etc. 

3.  Required  to  revert  the  general  series  x=ay'\-hy^-\' 
cf-^-dy*^  etc. 

Assume  y^Ax+Ex'^-^-Cx^-^-Dx*^  etc. 

ay=Aax+Bax^'\-Cax'^'\-Dac[^^  etc. 
Z>y2=         6^3a>2^25^j5^^ ( j52^ ^. 2ACh)x^,  etc. 
cy3=  c./73^+3^2j?ca7*,  etc.  ^  =0 

c;y*=  dA^x^^  etc. 


(art.  96,)^=-,    B=--,C=^—— 


D=- 


a 
5^3 — ^ahc^aH 


a? 
.    ,         X     hx^     263 — fljc         5^3 — 5a5c+a2c^ 

4.  Given 2r=w — -q-+-e- — ~7+"q"  +  >  ^*^'  *^  infinity,  to 
find  u  in  terms  of  z. 

Result,  ?i=2r+§2;3+/g2;5+^Yg2r7,  etc. 

«i2         1^3        7.4        «i5 

5.  Givena?=2^— ^  +  |- — T^r\ — j  etc.  to   infinity,   to 
find  y  in  terms  of  x. 

Result,  y:=^  +  -+-+^^  +  -^^^,  etc. 


128  SERIES. 


SUMMATION  OF  SERIES. 

103.  Infinite  series  are  sometimes  of  such  nature,  that 
a  quantity  can  be  found,  to  which  the  series  continually 
approximates,  and  to  which,  without  attaining  perfect 
equality,  it  arrives  more  nearly  than  by  any  assignable  dif- 
ference. The  quantity  to  which  the  series,  by  continued 
extension,  thus  approximates,  is  called  the  sum  of  the  in- 
finite series.  Thus  we  say,  .3333,  etc.  to  infinity,  =^5  for 
no  number  less  than  |  can  be  assigned,  which  the  series 
may  not,  by  extension,  be  made  to  exceed. 

Required  the  sum  of  l-{-x+x'^-{'X%  etc.  to  infinity,  x 
being  supposed  <^  1."^ 

Put  y—l+x-\-x^'{'X^,  etc.  to  infinity. 

.\  xy=x-\-x^+x'^j  etc.  to  infinity. 

Andy—xy=l.     .'.y^j^;^. 

2,  Required  the  sum  of  l  +  2x+3x^+4!X^'\-6x*+,  etc. 
to  infinity,  supposing  x<^  1. 

Put  2/=l+2a?+3a?2+4a?3+5a?*,  etc. 
...  — 2xy= — 2a?— 4073 — g^s — ga?*,  etc. 
And  x'^y=x^-\-2x^  +  3x\  etc. 
Adding  these  three  equations  together 
y — ^xy-^-x^y^l, 

_         1        _      1 

•'•  ^""  1— 2a7+a?3""(l— a?)2* 

3.  Required  the  sum  of  r^+"9~q"  +  o~4~5  ^^^'  *^  infinity. 

*The  character  <^  is  used  to  express  inequality,  the  opening  being 
presented  to  the  greater  quantity. 


SERIES.  129 

Assume  ^=l-4--rt-+-o--f-T-  +  "K"+j  ^tc.  to  infinity. 

Then  x-l=:-i+i-+-l+l+l,  etc. 

Subtract  the  latter  from  the  former,  and 
■^^lS "^273+31^ "^475'  ^*^'  *^^  ^""^  required. 
4.  Kequired  the  sum  of  jy3+2Ti"'"3X5'*^^^^"^^^"^' 
By  last  example,   l  =  j;2+2:3  +  3:5"^4:5'^*^* 
Subtract^.       ...^=^+^+^^+^,etc. 

By  subtraction,  i=.^3+^^+3|;5+j^^ 

Whence,  ^=ji-3  +  ^^  +  3^+ji^^     etc.  the  sum 

required. 

The  student  who  desires  to  pursue  this  subject,  may  con- 
sult Wood's  Algebra,  Article  411,  etc.,  or  Young's  Alge- 
bra, page  251  etc. 


DIFFERENTIAL  METHOD. 

104.  In  any  series  of  quantities,  «,  6,  c,  J,  e,  etc.,  if  each 
term  be  subtracted  from  the  next  following  one,  and  each 
term  of  the  series  of  differences  be  taken  from  the  next, 
and  so  on,  the  following  series  will  be  obtained. 

1st  differences,  h — a,  c — 5,  d — c,  e — <f,  etc. 
2d  diff.  c— 25+a,  cU-2c+6,  e— 2c?+c,  etc. 


130  SERIES. 

3d  difF.  (f— .3C+35— a,  e—3d+Sc—b,  etc. 

4th  diff.  e—4^d+6c—4^b+a,  etc. 

Or  these,  1st  diff.  — «+^,  — ^4-^,  — c+c?,  etc. 

2d  difF.  a — 25 +  c,  b—2c+d,  etc. 

3d  difF.  —a-\-3b—3c+d,  —b  +  3c—3d-^€,  etc. 

4th  difF.  a — ib  +  6c — 4c?+e, 

5th  diff.  —a  +  bb—10c^l0d—5e+f,  etc. 

By  a  little  attention  to  these  expressions,  and  a  com- 
parison of  their  formation  with  the  powers  of  a  binomial, 
we  perceive  that  if  .^  be  taken  to  denote  the  first  term  of 
any  (the  ^th)  order  of  differences. 

,     _  _        n — 1         n-->ln — 2_     , 

dtz^=^a — nb+n.—^c — n.         ,         d,  etc. 

the  sign  +  being  used  when  n  is  an  even  number,  and  — 
when  71  is  odd. 

If  the  differences  of  any  order  vanish,  any  one  of  the 
terms  may  be  found  by  means  of  the  others. 

Suppose  the  4th  difference  a — 45  + 6c — 4c^+e=0,  and  c 
was  not  known,  we  should  find, 

'-  6  ' 

105    Let  Z),  D,  D,  D,  etc.  denote  the  first  terms  of  the 

12        3  4 

1st,  2d,  3d,  4th,  etc.  orders  of  differences,  viz :  D=: — a+b, 

1 

D=a—2b'\'C,    JD= — a+3b—3c+d,    D=a^4<b-\-6c—4^d 

2  3  4 

+e,  etc.;  then  we  find  b=a-\'D,  c=a-\-2D  +  D^  d=a+ 

1  12 

3D  +  3I)+D,  c=a  +  4D+6D+4i)+D. 

12  8  12  8  4 


=*  By  this  method  the  computers  of  the  Nautical  Almanac  verify 
their  calculations  of  the  moon's  longitude,  latitude,  etc. 


SERIES.  131 

Hence,  the  n+lth  term  of  the  series 

^        n — 1  n — In — 2_ 

=a+wD+7i.-7c— D  +  w.-»— .— ^— I),  etc. 

1  -^2  Z  O       3 

Consequently  the  Tith  term 

^ — 2  ^ n — 2  n — 3 , 


=a+n—l.D+n—l.—^D  +n—l.-^.-^D,  etc. 

1.  Required  the  Tzth  term  of  the  series  of  odd  numbers, 
1,  3,  5,  7,  etc. 

•  Here  the  1st  diff.  are  2,  2,  etc. ;  2d  difF.  0. 


.•.the  nth  term  =l  +  2.n — 1  =2n — 1. 

106.  Required  the  sum  of  n  terms  of  the  series  a,  b,  c,  c?, 
etc. 

This  is  manifestly  the  same  as  the  n+lih  term  of  the 
following : 

0,  0+a,  0+a+b,  0+a+b  +  c,  etc. 
Hence,  by  the  last  article,  the  required  sum 

n — 1  _        n — 1  n — 2  ^        n — 1  n — 2  n — 3  ^ 

etc. 

1.  Required  the  sum  of  n  terms  of  the  square  numbers, 
1,  4,  9,  16,  etc. 

Here  a=l,  D=3,  D  =  2,  D=0. 

12  3 

.'.  the  sum 


,     n — 1^        n — In — 2^     n.n+l.^n+l 

2.  Required  the  sum  of  the  cube  numbers,  1,  8,  27,  64^, 
etc.  continued  to  n  terms. 

3.  Required  the  sum  of  25  terms  of  the  series,  1,  3,  5, 
7,  etc.  Ans.  625. 


...32  SERIES. 

4.  Required  the  sum  of  15  terms  of  the  series  1,  16,  81, 
256,  625.  1296,  ate.  Ans.  178312. 

When  the  differences  at  length  vanish,  any  term  of  the 
series,  or  the  sum  of  any  number  of  terms,  may  be  accu- 
rately determined  by  the  methods  used  in  this  and  the  pre- 
ceding articles :  when  the  differences  become  small,  but 
do  not  vanish,  a  near  approximation  can  be  made. 

107.  In  article  105,  the  number  n  is  supposed  to  be  an 
integer,  in  which  case,  if  the  differences  vanish,  the  rule  is 
demonstrably  correct  5  the  same  formula  is,  however,  ap- 
plied to  the  case  where  n  is  fractional. 

Suppose  a  series  ^,  ^,  r,  5,  etc.,  of  equi-different  quanti- 
ties, and  another  series,  a,  6,  c,  c/,  etc.,  such  that  the  terms 
of  the  latter  shall  be  similar  functions  of  the  correlative 
terms  of  the  former.  It  is  proposed  to  find  a  term  y,  in 
the  latter  series,  corresponding  to  v  in  the  former  5  v — p 
being  given. 

Let  q — p  :  v — p  : :  1 :  a?,  and  take  as  before,  /),  D,  D,  etc. 

13  3 

the  first  of  the  1st,  2d,  3d,  etc.  differences. 
Then  assuming  the  series,  (art.  105.) 

^        a: — 1  ^  0:^1  X — 2^     . 

y=a-\-3cB-\-x.—^D  +x.—^,—^D,  etc. 

This  gives,  when  a?=0,  y=a,  when  a?=  1,  y^=a-\-'D^  when 

1 

a?=2,  2/=a+2D  +  D.  etc.;  but  these  expressions  equal  Z>, 
1       3 

c,  etc.  as  they  ought  to  do.  This  formula,  being  demon- 
strably correct  when  x  is  integral,  is  assumed  as  a  conve- 
nient and  useful  approximation  when  a:  is  a  fraction. 

Required  to  find  the  6ith  term  in  the  series  1,  4,  9,  16. 

Herea:=-^  a=l,  D=3,  D=2,  D=0. 

-^  12  3 


LOGARITHMS.  '  133 

,     11     3     11     9     2     169 

2.  Le\  a,  b,  c,  c?,  be  arcs  of  a  great  circle  intercepted  be- 
tween a  fixed  star  and  the  moon's  centre  at  noon  and  mid- 
night of  two  successive  days,  it  is  required  to  find  the  dis- 
tance at  15  hours  from  the  first  noon. 

Here  12: 15  : :  1  :a?=^ 

4. 

...dist.  (3^)=«+|i)+|x^Z)  +  Jx^x7|l)= 


CONSTRUCTION  OF  LOGARITHMS. 

108.  From  article  92,  it  is  obvious  that)  a  being  any 

number,  if  a°=JV,  and  d^=M,  a^'^'^^J^M^  and  «''-"=  ^. 

If,  therefore,  all  the  numbers  used  in  calculation  were  ex- 
pressed in  powers  of  a,  multiplication  and  division  might 
be  performed  by  the  addition  and  subtraction  of  the  expo- 
nents of  a.  These  exponents,  thus  employed,  are  termed 
logarithms.     When  a°=^Jf^  n  is  the  logarithm  of  JV. 

Let  a?  be  such  number,  that  a^  may  denote  any  proposed 
number. 

Put  a=l-f5,  then  (art.  98,  99,  100,) 

X — 1  X — 1  X — 2  0?- — 1 

X — 2  X — 3.  ,  f  x'^    X  \ 

__.__6S  etc.     =  l+6x  +  6^  J  2 -2  S  + 

*i6-2+3i+*i2i— +^— l^*-^- 
12 


134?  LOGARITHMS. 

If,  therefore,  we  collect  the  terms  containing  the  succes- 
sive powers  of  x,  and  denote  the  above  equation  thus, 

(l-\-bY=l  +  ^x  +  Bx^+Cx^+Dx%  etc. 

We  readily  find  that 

^     ^     ¥     b^     ¥     h^ 

But  the  values  of  B,  C,  etc.  remain  to  be  determined. 

If  for  X  in  the  above  equation  we  substitute  2a?,  we  shall 
have 

(l4-i)2x=i+2^a7  +  45a?2+8Ca?3+16i)a?S  etc. 
Squaring  the  same  equation, 

(1  +  ^)3^=  l  +  2./^a?  +  25a72+2Ca?34.2Z)a?S  etc. 

A^x^'\'2ABx'+2ACx^,  etc. 

B^x\  etc. 

Comparing  the  co-efficients  of  like  powers  of  x. 

A^  A^  A^ 

^=^' ^=213-^=2X1' '**= 

A^x'^     A^x^     A*x* 
...a-=(l+6)'=l+^a?+-^  +  -^+2:3;-^,  etc.  (R.) 

In  like  manner, 

etc. 

A^x^    A^x^     A^x^  ^    .       , .  . 

=  1  4-^07  +  -^+-^  +2:3:4;'  ^^^-  ('^•)  '^  ^^'^^ 

52     b^     b^  ^ 

•^=^+2 +3-+4:'  ^^^-  (^) 

109.  Let  1  +  ^  denote  a  number  whose  logarithm  is  re- 
quired; then  l+2/=(l+^r' 


LOGARITHMS.  135 

.'.  (l'\-yY={l-hbY%  z  being  any  number  whatever. 
Here  x  is  the  logarithm  of  (1+b)'',  or  of  a%  or  1+y. 

But  (l+y)^=i  +  F2r+-^+-^,  etc.  (J?.  108.) 
Fbeing  ==y-|V|--?J  +  f ,  etc.  (Q.  108.) 

Also  (1+6)"=  1 4- ^^^H 2~"^T3~'  ^*^'  ^^'  ^^^'^ 

Whence,  by  comparing  the  co-efficients  of  2r, 
A.=  V,  and  ,^^^y-4y-+irzJy!±Ly-,  etc.  = 

y<2.  y^  y\  yS  J 

Again,  putting =  a  number  whose  logarithm  is 

required,  and  making  ^j =(14-6)^,  we  find 

V  (the  log.  of  (1  +5')  or  -^—)= 

-.3      t/2      y*      7® 
Jlfx  12/+%"+^  + 4  +  5  5  ^*^'  I  ^y  ^  process  exactly  ana- 
logous to  the  former. 

Now  the  logarithm  of  ■- (or  of  1  +  yX-— —  =  lo- 
garithm of  (1+2/)+  logarithm  of  ■— -  = 

(r+t;=2Jlf|y  +  ^+^+^,  etc.}  {U.) 

As  is  manifest  by  adding  together  the  values  of  x  and  v 
above  obtained. 


136  LOGARITHMS. 

110.  As  A^  and  consequently -^  or  JIf,  is  given  in  terms 
of  5,  {a — 1)  and  a  may  be  assumed  of  any  value  whatever, 
(art.  108j)  it  follows  that  M  is  not  limited  to  one  particu- 
lar value. 

If  the  value  of  a  be  assumed,  M  may  be  thence  deter- 
mined, or  if  we  take  any  number  at  pleasure  as  the  value 
of  JkT,*'  a  will  be  limited ;  but  in  this  case  the  formula  TJ 
may  be  applied  to  the  construction  of  logarithms,  without 
computing  the  value  of  a.  M  is  termed  the  modulus,  and 
a  the  radix  of  the  system.  Instead  of  fixing  the  value  of 
M^  or  G5,  by  arbitrary  assumption,  we  may  take  1  as  the 
logarithm  of  any  proposed  number,  and  thence  deduce  the 
value  of  M, 

Since  a^=l,  the  logarithm  of  1  must  in  every  system, 
=  0.  If  we  make  1=  logarithm  of  10,  the  radix  will  be 
10,  and  M  might  have  been  found  from  equation  /,  (art. 
108,)  by  assuming  6=9,  if  that  series  or  its  reciprocal  had 
been  a  converging  one.  As  that  is  not  the  case,  a  different 
expedient  must  be  used. 

jo+  1 
Let^^- -denote  a  number  whose  logarithm  is  to  be 

\^y     7?  +  l  1 

found,  and  put  - — -^= :  then  ^=p; — —r :  this  value 

'  ^      1 — y       p    '  ^     2p4-l 

substituted  for  y  in  equation  Z7,  (art.  109,)  will  make  the 

seri.es  always  converge,  and  more  rapidly  the  greater  the 

value  of  j9. 


*  When  Mis  assumed  =1,  the  logarithms  thence  obtained  are  termed 
Naperian,  from  th«  name  of  the  inventor,  or  hyperbolical,  from  their 
relation  to  the  areas  contained  between  the  curve  and  asymptote  of  an 
equilateral  hyperbola. 


LOGARITHMS.  137 

Now  since X  --=p-f-l,  it  is  evident  that  the  log.  of 

p  + 1=  log.  i?+ log.  of  -— . 

First,  let  p=l,  then =2,  and  -^ — -——-'  also  log. 

^=.3333333333 
-i=^=  370370370 

(|)'=^of  (|)3=  41152263 

(|)'  =  ^of  (g)5=  4572474(^0 


(3)^  =     6272  (C.) 

(J)«  =      697 

(|)"=)  77  (i).) 

12* 


138  LOGARITHMS, 

If  then,  we  divide  these  terms  by  the  co-efficients,  1,  3,  5, 
7,  etc.,  we  shall  have — 


3   ' 

.3333333333 

1 

3.33"" 

123456790 

1 
5.3^"" 

8230453 

1 

7.37~" 

653211 

1 

9.39"' 

56450 

1 
11.3^" 

5132 

1 
13.3^" 

483 

1 

15.3«" 

46 

1 
17.3^7- 

4 

.3465735902 

V- 


Hence,  log.  of  2=  .6931471804 Jf,  and  log.  of  8  (or  2^) 
=2.0794415412^1/:  J 

Second,  let  ;?=4,  then  ^^:^=^ 

5 

Hence  to  find  the  log.  of  ^ 


LOGARITHMS. 


139 


.1111111111 


g  of  {~f=^=     45724^74 


I  of  ilY=.6B. 


33870 


1  .  1    C 

7  ^^  (9) -21^     ^^^ 


.1115717757 


Whence,  log.  of -=223 14355 14 JIf. 
Consequently  the  log.  of  (tX  j)  10 


But  log.  of  10 
Hence,  M, 


=2.3025850926JJf. 
1. 
1 
2.3025850926 


,4342944822 


Substituting  for  M  the  number  last  found,  we  obtain  the 
log.  of  2=. 30102999599,  and  thence  the  log.  of  4,  8,  16, 
32,  etc.  may  be  had  by  simple  multiplication.     Also  the 
log.  of  5=  log.  of  10—  log.  of  2=. 69897000401;  whence' 
may  be  found  the  logarithms  of  all  the  powers  of  5. 

The  numerical  value  of  JIf  being  thus  ascertained,  if  we 
make  p  denote  a  number  whose  log.  is  known,  we  have 


log  p  +  1  =:log.j?+log.' 


p  +  1 


P 


:l0g.;7  + 


2^+1"^ 3.(2p  +  l)2 "^5.(2^  +  1)^+7.(2^  +  1)2' ^^^* 


140  LOGARITHMS. 

Where  Jl^  B,  C,  etc.  denote  the  preceding  term  exclu- 
sive of  the  divisor,  3,  5,  etc. ;  and  thus  the  logarithms  of 
all  the  prime  numbers  may  be  computed.  But  as  the  above 
series,  when  the  number  p  is  small,  converges  but  slowly, 
and  therefore  requires  a  considerable  number  of  terms  to 
be  used,  the  labor  may  be  abridged  by  proper  expedients, 
a  few  of  which  are  subjoined. 

Let  the  log.  of  3  be  required  j  putp=80,  then 


2p+l     161 


And  log.  of  ^=^+3^^921-         -0053950319 
To  which  add  log.  of  80=^1+3  log.  of  2==  1.9030899880 


Hence,  log.  of  81,  or  4  log.  of  3=  1.9084850199 

And  log.  3=. 4771212549. 

Again,  let  the  log.  of  7  be  required.     Put  J3=27,  then 

1___    1 

2^+1  ~^55 

28     2M      Jl 
And  log.  ^=:^  +^^         =  .0157942671 

Adding  log.  of  3^  or  of  27  =  1.4313637647 


Log.  of  28=  1.4471580318 
Subtracting  log  of  4,         .6020599920 


Log.  of  7  =  .8450980398 


*  As  the  logaritlims  of  10  and  its  powers  are  wholly  integral,  it  ia 
manifest  the  logarithnn  of  a  number  is  changed  only  in  the  integral 
part;  by  varying  the  position  of  the  decincial  point  in  the  number  itself^ 


LOGARITHMS.  141 

In  general,  suppose  a?4-2  to  denote  a  prime  number 
whose  logarithm  is  required,  those  of  the  inferior  numbers 
being  known. 

(x—lYx(x+2)     x^—Sx  +  'i       ,        ,     , 

femce  , ^-z.rr-1 rr,='^ — is ?»?  where  both  terms 

(a?  +  1)2  X  [x — 2)     x^ — 307—2 ' 

are  divisible  by  4,  because  x — 1,  and  07+1  are  even  num- 
bers, it  follows  that  this  fraction  in  its  lowest  terms,  may 

p+1 
be  expressed  by  ,  and  therefore  the  log.  of 

(07—1)2.(07+2) 

(a;+ 1)2.(07—2)' 
be  obtained  from  the  formulse  above  given ;  to  which  add- 
ing 2  log.  (o7+l)  +  log.  {x — 2) — 2  log.  (07 — 1)  the  result  will 
be  the  log.  of  07+2. 

Let  the  log.  of  13  be  required.  Here  07=11,  and  our 
fraction  becomes 

10U3_  1300  __  325 
T2"2;^'~1296~35i 

Hence,  .?.=324,  and  g^j=^ 

325 

Whence,  log.  ^^  =  .0013383507.  To  which  add  4  log. 

3+log.of  4-2,  logof5=. 1126050039 


Log.  of  13  =  1.1139433546 

In  this  manner  the  logarithms  may  be  derived  from 
those  already  obtained,  to  any  proposed  extent.  The  dif- 
ferential method  may  be  advantageously  applied  to  the 
completion  of  a  logarithmic  table.  But  the  labor  of  those 
computations,  being  already  finished,  and  not  likely  to  be 
renewed,  a  further  elucidation  of  the  subject  is  deemed 
unnecessary. 


142  SURDS. 

Section  X. 

SURDS. 

111.  It  has  been  remarked  (note,  art.  38,)  that  the  even 
roots  of  negative  quantities  are  impossible ;  hence  when- 
ever, in  the  solution  of  a  problem,  the  square'  root  of  a 
negative  quantity  appears  in  the  result,  such  result  is  im- 
possible, or  imaginary.  But  instead  of  abandoning,  as 
hopeless,  every  example  in  which  such  expressions  appear, 
they  are  found  conformable  to  the  general  principles  of 
the  science,  and  sometimes  connected  with  the  most  refined 
analytical  processes. 

Impossible  roots  may  be  introduced  into  a  problem  in 
two  ways,  quite  distinct  from  each  other.  First,  by  ad- 
mitting incompatible  assumptions  into  the  data  of  the  pro- 
blem ;  in  which  case  the  impossible  root  serves  to  detect 
that  incompatibility ;  and  its  appearance  or  disappearance 
marks  the  limits  of  the  problem. 

Thus  in  the  equation  x'^ — 2aa?= — 5,  we  have  (by  art. 
67,)  a?=azfc^/a2 — b  as  the  general  expression  of  the  value 
of  X,  If  now  a^^b,^  a^ — b  is  positive,  and  therefore, 
's/a^ — bj  and  consequently  x,  a  possible  quantity.  But 
if  a^<Cp^  ^^ — ^  is  negative;  and  therefore,  ^a^ — b  im- 
possible ;  whence  x  is,  in  this  case,  an  imaginary  quan- 
tity. Here  the  terms  of  the  original  equation  are  com- 
patible, whenever  a^  is  equal  to,  or  greater  than  b;  but 
incompatible  when  a^<^b  ;  for  x^ — 2ax= — b  is  equivalent 
to  2a — x.x=b,  but  2« — x.x  cannot  be  greater  than  a^. 


Put  x=aztc,  then  2a — x=azpc,  and  2a — x.x=a^ — c^. 

When  c=0,  x=a,  or  2a — x.x=(b=^a^'  this  is  there- 
fore the  limit  of  b, 

*a'^b  is  read  a  is  greater  than  h,  and  a<Zb,  and  a  is  less  than  b. 


SURDS.  143 

Suppose  a?+-  =  2«,  to  find  a?. 

Multiplying  by  a?,  a?^4-l  =  2«a?,  or  o?^ — 2«a?= — 1. 


Whence,  x=azri  v/a^ — 1,  which  value  of  a?  is  imaginary 
when  «<^1.  Hence  the  least  possible  sum  of  a  number 
and  its  reciprocal  is  2.*" 

112.  Impossible  roots  may  be  sometimes  obtained,  when 
the  data  are  compatible,  by  the  admission  of  inconsistent 
suppositions  into  the  solution. 

Suppose  we  have  x'^-^ny'^=a,  and  xy=h,  to  find  x  and 
y ;  a,  b,  and  n  being  any  given  numbers. 

To  and  from  the  first  equation  add  and  subtract  2^/71 
times  the  second,  whence,  x'^'{-2xy\/n-{-ny^=a  +  2by/ny 
and  x'^ — ^xy^/n-^ny^^a — 2by/n;  and  by  evolution, 


x-^ys/n—s/  a-Y'lbs/n^  (M.) 


X — y  s/n=s/ a — 2b  s/ n^  ( JV.) 


Va  +  2Z>v'^+N/« — 2by/7i 
.x= :^ 


*  These  principles  may  be  applied  to  determine  the  maxima 
and  minima  of  geometrical  quantities.  Let  b  =  the  base,  «  =  the 
altitude  of  a  plane  triangle,  x  =  altitude  of  its  inscribed  rectangle ; 

thenbx =    the     area    of    the    rectangle,     which    put    =c  ; 

a 

whence    we    find   x=-zi^K/  ('- ),    which  is  impossible    when 

2  4b 

^ab 

If  c——,  x=z~,  this  is,  therefore,  the  greatest  possible  value  of  x, 
4  2 

and  the  greatest  rectangle  is  — ^  -  the  triangle. 

4r  2 


14i^  SURDS. 


And  y= — ^c—. 

Which  are  general  expressions  for  the  values  of  a?  and  y. 

If  now  we  take  n= — 1,  or  a?^ — y^=a,  the  above  expres- 
sions become 


X— ^ 7Z ■ 


N/a+26v/— 1— \/a— 26v'  — 1 
2/= 27=1 

These  values,  though  expressed  by  imaginary  quantities, 
are  real  ones.  For  putting  — 1  instead  of  n^  in  equations 
JkT,  JV,  and  multiplying,  we  have 


a?2+y3_^^3_|.4^3^  (P.) 
But  from  the  first  equation,  a?^ — y^=.a. 
Consequently,  by  addition  and  subtraction, 


2x''=y/a^-\-^b^-\'a. 


And  2y2=v/aH4^^'— «, 


ora^=:( g f;    y=( g ) 

The  same  conclusions  may  be  obtained  by  squaring  the 
values  of  x  and  y  first  found,  and  extracting  the  square 
roots  of  the  results. 

The  impossible  surd  V — 1,  was  evidently  introduced 
into  the  result,  by  adopting  a  process  in  the  general  solu- 
tion, which  was  not  applicable  to  the  particular  equation 
x^ — y'^—a;  yet  the  values  of  x  and  y,  when  cleared  of 
imaginary  surds,  are  the  true  ones ,  as  may  be  shown  by 
a  different  solution  of  the  problem. 


SURDS.  145 

To  the  square  of  the  first  equation  adding  4  times  the 
square  of  the  second,  x^-\-2x'^y^'^y~=^a'^  +  4:b^. 

Whence,  by  evolution,  x'^-\-y^=  y/a^-{-4^b^j  the  same  as 
equation  P,  deduced  from  imaginary  surds. 

113.  From  what  is  shown  in  the  foregoing  article,  we 
readily  infer,  that  when,  in  the  solution  of  a  problem,  the 
value  of  the  quantity  sought  appears  in  terms  of  imaginary 
surds,  we  are  not  thence  immediately  to  conclude,  that  the 
data  are  inconsistent ;  as  the  adoption  of  an  inapplicable 
process  may  produce  such  a  result ;  yet  in  this  case  the 
imaginary  surd  may  be  eliminated  by  the  use  of  proper 
expedients.  When,  however,  the  data  are  inconsistent,  no 
analytical  address  can  clear  the  final  equation  of  its  im- 
possible quantities. 

Imaginary  surds  differ  from  real  ones  in  this  important 
particular.  Keal  surds,  however  complex,  admit  of  an 
approximation  to  their  value ;  but  imaginary  surds  admit 
of  no  approximation,  and  must  either  be  eliminated,  when 
practicable,  or  remain  the  intractable  indications  of  incon- 
gruous assumptions. 

The  following  cases  exhibit  the  most  useful  applications 
of  algebra  to  surd  quantities. 

Case  I. 

114.  To  reduce  a  rational  quantity  to  the  form  of  a  surd. 

Raise  the  given  quantity  to  the  power  denoted  by  the 
index  of  the  surd,  and  to  this  power  apply  the  radical  sign 
or  index  proposed. 

EXAMPLES. 

1.  Reduce  5  to  the  form  of  a  square  root,  and  a  to  that 
of  a  4th  root. 

5=v^5-=v/25,  and  a=(a^y=i/a\ 

2.  Reduce  3  to  the  form  of  a  cube  root. 

Result,  -727. 
13 


146  SURDS. 

3.  Express  — -^a  in  the  form  of  a  cube  root. 

o 

I         A. 

Result,  (—^a^)'. 

4.  Reduce  2\/5  to  the  form  of  a  square  root. 

Result,  v'20. 

5.  Reduce  3  \/  2  to  the  form  of  a  ^th  root. 

Result,  (324)*. 

6.  Express  a-^b  in  the  form  of  a  square  root. 


7.  Express  — ^ —  in  the  form  of  a  quadratic  surd 


Result,  v/a'^+2a6  +  63. 

2  surd. 
6+2v/5 


Result,  s/  -      . 

Case  2. 

115.  To  reduce  radical  quantities^  having  different  in- 
dices^ to  other  equivalent  quantities  with  a  common 
radical  sign. 

Reduce  the  fractional  exponents  to  a  common  deno- 
minator, involve  the  given  quantities  to  the  powers  de- 
noted by  their  respective  numerators ;  and  to  the  results 
apply  the  reciprocal  of  the  common  denominator  as  the 
common  exponent. 

EXAMPLES. 
2         3  1 

1.  Reduce  a^^  b"^,  and  c^  to  equivalent  quantities, 
having  a  common  exponent. 

2     3     1^      ^  .     ,     _    40    45     12 

3,  -,  g  reduced  are  equivalent  to  g^,  g^,  - 

2  1  4C 


Hence,  a^  =  (a'^)^^=a^^', 

3  1        45      1  1        12 


SURDS.  147 

1  JL 

2    Keduce  3  ,  and  V  to  a  common  mdex. 

Eesult,  27^,  16^. 

3.  Keduce  a^,  b^,  to  a  common  radical  sign. 

Result,  aT\  b'f~5. 

4.  Reduce  {a-\-x)^,  (a — x)^  to  a  common  index. 
Result,  {a''+2ax-{'X^)^,  and  (a^ — 3a^x-^3ax^—x^)^. 

IX  1 

5.  Reduce  a^,  and  a?*  to  the  common  index  —.* 

I/O 

Result,  (a*)i"5,  (a?3)T2. 

1  i 

6.  Reduce  4^,  and  5"*  to  a  common  index. 

Result,  ( 16)7^2  and  ( 125)^5. , 

Case  3 
116.  To  reduce  surds  to  their  most  simple  terms. 

Resolve  the  surd,  if  possible,  into  two  factors,  one  of 
which  shall  be  the  greatest  power  it  contains.  Extract  the 
root,  and  thereto  annex  the  other  factor  with  its  proper 
radical  sign. 

EXAMPLES. 

1.  Reduce  V250  to  its  simplest  terms. 

250:^125X2=5^^X2.     .-.  ^250=5^2. 

2.  Reduce   x/32  to  its  simplest  terms. 

Result,  4  v^  2. 


*  When  the  common  index,  to  which  the  fractional  exponents  are  to 
be  reduced,  is  given,  divide  each  of  the  given  exponents  by  that  com- 
mon index,  and  involve  the  given  quantities  to  the  powers  indicated 
by  the  quotients. 


148  SURDS.  ^ 

3.  Reduce  V243  to  its  simplest  form. 

Result,  3 Vs. 

4.  Reduce  V^IS^  to  its  simplest  terms. 

Result,  12  V  3. 

5.  Reduce  (144)*  to  its  simplest  terms. 

Result,  2  V  3. 

6.  Reduce  [a^b — a^x)^  to  its  most  simple  terms. 

Result,  al/b — x, 

18 

7.  Reduce    \^t^  to  its  simplest  terms.^ 


3 
Result,  ^VIO. 


8.  Reduce    V  5  to  its  simplest  terms. 


9 


Result,  I  Vs. 


9.    Reduce   — -^ j^-  to  its  simplest  form. 

Result,  8+2v/ 15. 

Case  4. 

117.  To  add  or  subtract  surd  quantities. 

Reduce  the  quantities,  when  fractional,  to  a  common 
denominator;  and  when  the  exponents  are  different,  to  a 
common  radical  sign.  Also  express  the  surds  in  their 
simplest  terms.  If  then  the  reduced  surds  are  alike,  they 
may  be  added  or  subtracted  as  other  algebraic  quantities. 


*  When  the  given  surd  is  fractional,  the  denominator  nnay  generally 
be  made  rational,  without  changing  the  value  of  the  fraction,  by  mul- 
tiplying both  terms  by  proper  numbers  or  quantities.  When  the  de- 
nominator consists  of  two  quadratic  surds  connected  by  the  sign  +  or 
— 5  the  proper  multiplier  consists  of  the  same  surds  connected  by  the 
opposite  sign. 


SURDS.  149 

When  the  surds  are  unlike,  the  operation  must  be  merely 
indicated. 


1.  Required  the  sum  and  difference  of  1/ 128  and  V 16. 

V128=4V2,  V  16  =  2^2,  4V2+2V2=6V2, 
4^2— 2V2=2V2. 

2.  What  is  the  sum  of  ^/27  and  >/48'? 

Ans.  Tv/S. 

3.  What  is  the  difference  between  \/50  and  v/18'? 

Ans.  2v/2. 

4.  Required  the  sum  of  ^56  and  Vl89. 

Ans.  5V7. 

5.  What  is  the  sum  oi  ^^/a'^b  and  5  V  16a*5 '? 

Ans.  (3a +20^2)  ^5. 

6.  What   difference   is   there   between    ^2^a^h^    and 
V  54a6*  1  Ans.  (2a6— 36^)  ^Z  6a. 


7.  Required  the  sum  of  Vt  and  V  09  '^ 


Ans.  J  V  2. 


^/14+^/12 
8.  Required  the  difference  between  — j^ j-^  and 

^3~^^^  Ans.  872+4.x/21  +  4x/3. 

Case  5. 
118.   To  multiply  or  divide  surd  quantities. 

Like  quantities,  with  different  exponents,  are  multiplied 
or  divided  by  adding  or  subtracting  their  indices. 
13* 


150  SURDS. 

When  the  quantities  are  different  reduce  them,  if  neces- 
sary, to  their  equivalent  ones  with  a  common  radical  sign. 
Then  the  product  or  quotient,  with  a  common  radical  sign 
applied,  will  be  the  quantity  sought.* 

EXAMPLES. 

1.  Eequired  the  product  of  5V2  by  7^3. 

5V2=5V4,  7v/3=7V27,  5  V^^X  7V27=r:35Vl08. 

2.  Divide  x/21+v/15  by  ^7— v/5. 

x/21+n/15_ x/21  +  \/15       V7+v/5  _  12v/3  +  2v/105 
v^7_^5~-    ^7—^5    ^  V7+v/5  ~  7—5 

=  6v/3+'>/105,  the  quotient  required. 

3.  Required  the  product  of  5  v/ 2  by  7  V  3. 

Result,  35  v/ 6. 

4.  What  is  the  product  of  3  v/ 5  by  4V25  % 

Ans,  60. 

5     3         9      2 

5.  What  is  the  product  of^v/^  by  tkn/c'? 

^^^•20^^^- 

6.  What  is  the  product  of  2V  U  and  3  V^^  '^ 

Ans.  12  V  7. 

7.  What  is  the  quotient  of  6^972  by  31/21 

Ans.  6Vl8. 


•  In  the  division  of  surds,  as  well  as  of  rational  quantities,  it  is  fre- 
quently convenient  to  set  down  the  terms  as  a  vulgar  fraction,  and  to 
reduce  that  fraction  to  its  simplest  forijp. 


SURDS.  151 

gs/^  divided  by  |v/g 


5      1  3     2 

8.  What  quotient  will  -fV^  divided  by  -VT  make  % 


Ans.QV5. 

9.  What  is  the  product  of  (a+b)^  and  (a  +  b)^  1 

Ans.  (a+byk 


10.  What  is  the  product  of{a  +  2Vb)^  and  (a—^Vby^l 

Ans.  (a^ — 4.6)2". 

11.  What  is  the  product  of  3+  v" — -2  by  5— x/— 2'? 

Ans.  17  +  2V— 2 

12.  What  is  the  product  of  (a  +  V—b)^  by  (a--'2  s/ —b)i  1 

Ans.  (a3+2&— av/— 5)i 

4  2  6  ^ 

13.  Divide  ^Vfl^ by- V«'  Quotient, -a^. 

XX  3  ^r? 

14.  Divide  3a  °  by  ^a"'  Quot.  ^a  ™  ' 

15.  Divide  5+3^^—3  by  7— 2v/— 3. 

17     31 
Quot.g^+g3V-3. 

As  involution  is  effected  by  the  multiplication  of  equal 
factors,  this  case  evidently  includes  the  involution  of  surd 
quantities. 

16.  What  is  the  3d  power  of  4  v"  2 1 

Ans.  128  v/ 2. 

X  n 

17.  Required  the  nth  power  of  a""  1  Ans.  am, 

18.  What  is  the  3d  power  of  3+2n/— 1 1 

Ans.  46  x/— 1—9. 


152  SURDS. 

Case  6. 

119.  To  extract  the  roots  of  surd  quantities. 

When  the  quantity  consists  of  one  term  only,  the  root 
is  obtained  by  dividing  the  index  of  each  quantity  by  the 
index  of  the  power. 

When  the  quantity  is  a  compound  one,  the  root  may 
sometimes  be  extracted,  as  in  article  39. 

When  the  given  quantity  is  a  binomial,  one  term,  at 
least,  of  which  is  a  quadratic  surd,  the  square  root  may 
sometimes  be  extracted  by  the  following  formula. 


Let  y/adt:y/b=^x,  put  s/a^ — 6=c?,  then 


*  This  formula  is  thus  obtained. 


Let  \/a-±:'^h—'sJv:^s/w, 
Then  a±is/h=v±i2s/vW'\-w, 

Now,  as  a  rational  quantity  cannot  be  equal  to  a  surd,  we  must 
take 

a=v+w^  (Jf.)  and  s/h—^s/vw. 
By  squaring  these  equations, 

a2=2;3^2vw+wj2,  and  b=4tvw. 
Whence,  by  subtraction  and  evolution, 


V  «^ — b  =  s/  v^ — 2vw  -^w^y  or  d=  v — w. 
From  this  and  equation  N,  we  have 

v=o^  4  o«)  a^d  w=-^a — -^d. 

X—  y/vdzs/wz=  ^ (-a  +  ^d)  ±:y/  (^a  —  -d.) 

Hence,  it  appears  that  this  method  applies  only  to  the  case  where 
a^ — h  is  a  complete  square. 


SURDS.  153 

EXAMPLES. 

1.  Eequired  the  square  root  of  16^/6, 

^/16=4,  and  ^6^=6^,     .-.  ^1l67Q=4<\/6. 

2.  What  is  the  square  root  of  6^  1 

Ans.  6^  or  6v^6. 


3.  What  is  the  cube  root  of  ^x/SI 


Ans.gV3 


4.  What  is  the  square  root  of  a^ — 6a  v/ — b — 9^? 

Ans.  a — 3\/ — b, 

5.  Required  the  square  root  of  6+2\/5. 

Herea=6,  n/6=:2v'5=  V20. 

.'.b=20,  and  ^=v/ 36— 20=4. 

1         6+4         6 — 4 
Whence  (6  +  2v/5)^=  V'-^^  N/-y- =  v^  5  +  1,  the 

root  required. 

6.  Whatisthesquareroot  of  3— 2v/2'? 

Ans.  V2— 1. 

7.  What  is  the  square  root  of  7— 2n/  10 '? 

Ans.  v'5— v'2. 

PROMISCUOUS  EXAMPLES. 

120.  Required  the  difference  between 


V5+2n/5         3+n/5  n/5  +  2v/5 

2""      ^'''^2V(5  +  '2V5)'  2        ' 


154  SURDS. 

5  +  2V5  5  +  2v/5  3  +  v/5 


2v/(5  +  2n/5)'  2v^(5+2n/5)     2v/(5  +  2v/5)" 


2v/(5+2V5r2^/(5+2v/5r       ^^^^^^^^^^^^  ^^^-  ^^^ 


V5+2V5     1 


denom.  by  ^/5-2^/5)  ^  ^^^"    =oV^  (l  +  5>/5) 
the  difference  sought. 

372+2 

2.  Kequired  the  sum  of  5^2 — 1,  and  ^ ^  y-^  1 

Ans.  8^2  +  3. 

3.  What  is  the  sum  of  12V  j  and  ^V'^'l 

27 
Ans.  "-tn/J^. 

4i.  Required  the  difference  of  3V3  and  X/121 

Ans.  v/9. 

5.  What  difference  is  there  between  2  n/ a^6*  and  a/  Sa^i^  1 

Ans.  2ab(a—b)l/b. 

6.  Required  the  product  of  4 + 2  V  2  by  2—  v/  2 '? 

Ans.  4.    • 

7.  Multiply  a+bs/—l  by  a— 5s/— 1. 

Product,  a2+53. 

44- V— 3  ,      1— n/— 3 

8.  Divide  —^-^ by  ^ . 

.  l  +  5v/— 3 

Quotient, j^ . 

9.  Required  the  square  root  of  51 — lOv/21 

Ans.  5^2— 1. 


EQUATIONS  IN  GENERAL.  155 

10.  Required  the  3d  power  of  a — 6y—  1  % 

Axis,  a?^— 3a2^V — 1 — SaS^+^V — 1. 

11.  Required  the  square  root  of  7 — 24x/ — 1  \ 

^  Ans.  4— 3v/— 1. 

12.  Divide  a?'\-h^  by  a—hsf—X^ 

Quotient,  a-^hsf — 1. 

13.  Add  —-7—; — 17  to r-, r. 

2a3— 2^2 
Sum,  — T-nir* 

\/7  a/  3 

14.  Add  -^^-^and-^-^--^ 

Sum,  g. 

15.  Divide  18  +  26V— Iby  3  +  >/— 1. 

Quot.  8  +  6  y—l. 

16.  Required  the  square  root  of  13 — 20  x/ — 3  % 

Ans.  5— 2v/— 3. 

Imaginary  surds  are  of  great  importance,  in  the  investi- 
gation of  several  valuable  formulss,  in  the  arithmetic  of 
sinesc 


Section  XI. 
EQUATIONS  IN  GENERAL. 

121.  It  has  been  remarked,  page  90,  that  quadratic 
equations  sometimes  admit  of  more  answers  than  one. 
The  principles  on  which  the  ambiguity  of  quadratic  equa- 
tions depends,  are  productive  of  similar  results  in  equations 
of  the  higher  orders. 


156  EQUATIONS    IN    GENERAL. 

In   the   general  -equation    a?° — •j9a?''~^+5'a?"~^......dbr=0, 

where  p^  q,  r,  are  supposed  to  be  given,  the  different  values, 
of  which  X  is  susceptible,  are  called  the  roots  of  the  equa- 
tion. 

122.  If  two  binomials,  as  x — a.  a?-W,  be  multiplied  to- 
gether, the  product  x"^ — ax — bx-^ab,  or  a?^ — (a-^b)x-{-ab,  is 
manifestly  a  quadratic;  if  now,  x — a— Of  or  x — ^=0,  we 
have  evidently  0?'^ — (a-\-b)x+ab—0.  On  the  other  hand, 
the  given  quadratic  equation  x^ — px-\-q—0,  may  be  re- 
solved by  making  a-{-b—p^  ab=q^  and  determining  a  and 
b.  For,  on  these  assumptions  being  made,  the  equations 
become  identical,  and  their  conditions  are  fulfilled  by 
taking  a?=a,  or  x=b;  the  equation  x" — px-\-q=0,  has, 
therefore,  two  positive  roots,  which  are  both  possible  when 
{p^  is  greater  than  q^  and  both  impossible  when  q  exceeds 
i/>2.     This  expression  corresponds  to  form  3d,  ^,  (art.  67.) 


^  123.  Again,  a? — axx-^b=x^ — ax-i-bx — a5,  which  being 
supposed  =0,  corresponds  to  x'^-i-px — ^'=0,  or  to  x'^ — px 
— ^=0,  according  as  b  is  greater  or  less  than  a. 

The  conditions  of  this  equation  are  answered  by  taking 
X — a=0^  or  x-{-b=0;  the  equation  has,  therefore,  two 
roots,  a,  and  —b.  These  expressions  correspond  to  forms 
1,  2,  c/f,  (art.  67.)  These  equations  consequently  admit  of 
a  negative  and  a  positive  root,  which  are  both  possible  ;  be- 
cause when  the  square  is  completed,  the  second  member 
consists  of  positive  quantities. 

124^  Moreover,  {x-\-a),{x-^b)=x"-{-ax-{-bx-{-ab,  which 
being  supposed  =0,  agrees  with  x'^-\-px-{-q=^0. 

Hence  an  equation  of  this  form  is  resolved  by  making 
a-\-b=p^  ab=zq^  and  determining  a  and  b.  The  roots  of 
this  equation  are  manifestly  both  negative;  and  (as  in  art. 
122,)  both  possible  or  both  impossible. 

125.  It  is  worthy  of  remark,  that  in  the  equation 
a?2 — px+q=::0^  the  signs  are  alternately  -|-  and  — ,  that  is, 
they  are  twice  changed ;  and  the  equation  has  tivo  positive 
roots.     But  in  the  equations  (art.  123,)  the  same  sign  is 


EQUATIONS   IN   GENERAL.  157 

once  continued,  and  the  sign  once  changed  ;  and  these  equa- 
tions have  one  affirmative  and  one  negative  root.  In  the 
equation  (art.  124,)  the  sign  is  twice  continued^  and  this 
equation  has  two  negative  roots. 

It  also  appears,  from  what  is  above  shown,  that  every 
quadratic  equation  has  two  roots,  which  are  both  possible, 
or  both  impossible.* 

126.  Assuming  three  factors,  (a? — a).(x — b).(x — c)= 
x'^ — (a-i-b  +  c)x''^+(ab'^ac  +  bc)x — abc,  which  being  sup- 
posed =  0,  will  be  identical  with  a?^ — px'^  +  qx — r=0,  if 
p  =  a-{'b  -{-c^  q=ab-\-ac-j'bc^  and  r=abc.  But  the  condi- 
tions of  the  former  are  answered  by  making  x — a=0, 
X — 6=0,  or  X — c=0;  therefore,  the  latter  has  three  posi- 
tive roots,  a,  bj  and  c.  It  has  likewise  three  changes  of  the 
signs. 

127.  If  instead  of  x — c  we  take  a?+c,  our  equation 
(x — a).(x — b).(x-\^c)  =  x'^' — (a  4-5 — c)x''^  +  (ab — ac — 6c)a?  + 
abc=0,  will,  manifestly,  have  two  positive  roots,  a  and  b, 
and  one  negative  root,  — c.  In  this  case,  if  a+b'^c,  the 
sign  of  the  second  term  is  — ,  the  equation  may,  therefore, 
be  expressed  x^ — px^dzqx-^r=0,  in  which  there  are  two 
changes  of  the  signs,  and  one  continuation  of  the  same  sign. 
If  c^a4-6,  the  second  term  will  have  the  sign  +,  but  the 
third  — ,  because  in  that  case  {a-\-b),  c^{a-^bY,  and 
therefore,  ^ab.  The  equation  then  becomes  x'^-^px^ — 
qx-{-r=0 ;  having  as  before  two  changes  of  the  signs,  and 
one  continuation  of  the  same  sign. 

128.  A  cubic  equation,  composed  of  the  factors  (x — a), 
(a? + 6) .  (a?  +  c)  =  o?"^  +  (6 + c — a)x'^ — (a6.4-  ac — bc)x — abc=  0, 
has  plainly  one  positive  root,  «,  and  two  negative  roots, 
— &,  and  — c.  But  in  this  case,  if  6  +  6-^a,  the  sign  of 
the  second  term  is  +5  and  the  equation  may  be  expressed 
x^-^px^ztzqx — r=0,  having  one  change  of  the  signs,  and 

*In  form  3,  A,  (art.  67,)  when  h=a^,  the  roots  are  equal  to  each 
other.  In  forms  1  and  2,  if  2a=0,  the  equation  becomes  a  simple 
quadratic,  and  the  roots  are  equal  quantities  with  contrary  signs. 
Every  simple  quadratic  has,  therefore,  a  positive  and  negative  root. 

u 


158  EQUATIONS   IN    GENERAL. 

two  continuations  of  the  same  sign.  .  If  a^h-\-c^  the  second 
and  third  terms  are  both  negative,  because  (h-\'C)d^(b-\-cf 
^bc;  the  equation,  therefore,  may  be  expressed  x'^ — px'^ — 
qx — r=0,  having,  as  before,  one  change  of  the  signs,  and 
two  continuations  of  the  same  sign. 

129.  If  we  use  the  factors  x-\-a,  x-\-b,  ^+c,  their  pro- 
duct a?^+(«-|'6-fc)a?'2+(a6+ac+Z>c)a7+a5c,  being  put  —0, 
the  equation  may  be  expressed,  a?^+j9^-+ga?4-r— 0,  in 
which  there  are  three  continuations  of  the  same  sign. 
The  equation  has  likewise  three  negative  roots,  — a,  — 6, 
and  — c. 

By  pursuing  this  inquiry  it  will  be  found,  that  any 
equation  of  this  kind  admits  of  as  many  roots  as  there  are 
units  in  the  index  of  the  highest  power  of  the  unknown 
quantity ;  that  the  number  of  positive  and  negative  roots 
will  be,  respectively,  equal  to  the  number  of  changes  in 
the  signs,  and  the  number  of  continuations  of  the  same 
sign.  It  likewise  appears,  that  the  last  term,  or  absolute 
number,  is  the  continued  product  of  all  the  roots  with  their 
signs  changed. 

130.  It  may  be  observed,  that  as  every  cubic  equation 
is  composed  of  three  factors,  and  every  quadratic  of  two ; 
a  cubic  equation  may  always  be  considered  as  the  product 
of  a  simple  and  a  quadratic  equation.  But  (art.  125,) 
every  quadratic  has,  either  two  possible,  or  two  impossible 
roots ;  hence,  a  cubic  equation,  the  terms  of  which  are 
possible,  having  one  impossible  root,  has  two;  and,  as  in 
the  multiplication  of  compound  quantities,  containing  im- 
possible parts,  those  impossible  parts  can  disappear  only 
when  two  like  roots  are  multiplied  together,  it  follows, 
that  every  cubic  equation,  consisting  of  possible  quantities, 
has,  at  least,  one  possible  root. 

131.  In  like  manner,  it  appears  that  every  equation  con- 
sisting of  possible  quantities,  having  an  odd  number  of 
roots,  has,  at  least,  one  of  those  roots  possible.     And  that 


EQUATIONS  IN  GENERAL.  159 

every  equation  which  is  made  up  of  possible  quantities, 
has  all  its  roots  possible,  or  an  even  number  of  impossible 
roots. 

132.  When  the  roots  of  an  equation  are  integral,  they 
may  sometimes  be  found  with  great  facility,  by  seeking 
the  divisors  of  the  last  term,  and  substituting  them  in  place 
of  the  unknown  quantity,  till  one  or  m.ore  are  found  which 
answer  the  conditions  of  the  equation.     (See  art.  129.) 

When  one  root  has  been  found,  the  equation  may  be 
depressed  by  connecting  that  root,  with  its  sign  changed, 
with  the  unknown  quantity,  and  dividing  the  given  equa- 
tion by  the  sum, 

EXAMPLES. 

1.  Given  x^ — Saj^-f  5a? — 15  =  0,  to  find  the  value  of  a?. 
(Art.  126,)  the  roots,  if  possible,  are  all  positive.* 
Also,  the  divisors  of  15  are  1,  3,  5,  15. 

Now,  by  substituting  these  for  a:, 

1_3.|-5_15^_-12.     ...  1  is  not  a  root. 

27— 27-f  15—15=0.     .-.  3  is  a  root. 

«     ^        ,  0?^— 3a?24-5a;— 15  ^     ^ 

a?— 3  =  0,  and  —, =a?2+5=0. 

X'      o 

Whence,  a:=zt:V  —  5. 

.-.  the  roots  are  3,  ■+  -s/ — 5,  and  — v/ — 5. 

2.  Given  a?^— 2a?2— 5a7+6  =  0,  to  find  x. 

Eesults,  1,  3,  —-2. 


*  The  rules,  in  the  foregoing  articles,  for  determining  the  signs  of 
the  roots  from  the  changes  in  the  signs  of  the  terms  composing  the 
equation,  being  founded  on  the  supposition  that  each  root  has  but  one 
sign,  do  not  apply  to  impossible  roots ;  because  the  negative  signs 
under  the  radicals,  when  developed  by  multiplication,  are  combined 
with  those  of  the  roots,  and,  therefore,  change  the  signs  of  the  terms 
of  which  the  equation  is  composed. 


160  EQUATIONS  IN  GENERAL. 

3.  Given  a?*^+6a?2— 7a?— 60=0,  to  find  x 

Results,  3,  — 4,  — 5 

4.  Given  a?^+3a:'-— 6a?— 8=0,  to  find  a?. 

Results,  2,  — 1,  — 4-. 

5.  Given  a?^— 2a?+4=0,  to  find  x. 

Results, —2,  l+\/— 1,  l--v/— -1. 

6.  Given  a?*— 10a?3  +  35a?^— 50a7+24=0,  to  find  x. 

Results,  1,  2,  3,  4. 

7.  Given  a?*-— So'H  ^Sa?^— 64a? +120=0,  to  find  a?. 

Results,  5,  3,  2^—2,  — 2n/— 2. 

133.  When  the  roots  are  not  integral,  they  may  gene- 
rally be  determined  by  approximation.*  For  this  purpose, 
various  rules  have  been  investigated.  Among  these  the 
following  is  probably  the  most  convenient  in  practice. 
The  demonstration  is  given  in  the  subsequent  article. 

To  find  the  root  of  a  general  equation. 

1.  If  all  the  terms  of  the  equation  are  not  on  one  side, 
by  transposition,  place  them  so  j  and  arrange  them  accord- 
ing to  the  powers  of  the  unknown  quantity,  placing  the 
highest  power  on  the  left  hand.  If  any  of  the  lower 
powers  are  not  contained  in  the  equation,  consider  each 
one  omitted,  as  having  a  cipher  for  its  co-efficient, 

2.  Place  the  co-efiicients  and  the  absolute  number  with 
their  proper  signs,  in  order,  in  a  horizontal  line. 


*If  all  the  roots  of  the  equation  are  impossible,  this  method  is  not 
applicable,  (see  art.  Ill,)  but  possible  roots  may  always  be  approxi- 
mated. 


EQUATIONS  IN  GENERAL.  161 

3.  Find  by  trial  the  first  root  figure,  attending  to  its 
value  as  being  units,  tens,  tenths,  or  hundredths,  etc  and 
place  it  to  the  right  of  the  absolute  number. 

4.  Multiply  the  first  co-efficient  by  the  root  figure,  and 
add  the  product  to  the  second  co-efficient ;  multiply  the 
sum  by  the  root  figure,  and  add  the  product  to  the  third 
co-efficient ;  proceed  thus  to  the  end  of  the  line,  adding 
the  last  product  to  the  absolute  number.  Again,  multiply 
the  first  co-efficient  by  the  root  figure,  and  add  the  pro- 
duct to  the  sum  under  the  second  co-efficient ;  multiply 
the  resulting  sum  by  the  root  figure,  and  add  the  pro- 
duct to  the  sum  under  the  third  co-efficient ',  and  so  on, 
stopping  under  the  last  co-efficient.  Repeat  the  process, 
stopping  each  succeeding  time,  one  term  nearer  to  the 
left  hand,  till  the  last  sum  falls  under  the  second  co- 
efficient. 

5.  Try  how  often  the  last  sum  under  the  last  co-effieient 
is  contained  in  the  sum-  under  the  absolute  number,  and 
take  the  result  for  the  next  root  figure. 

6.  Using  the  first  co-efficient,  and  the  last  sum  in  each 
column,  instead  of  the  co-efficients  and  absolute  number, 
proceed  with  this  new  root  figure  as  with  the  preceding 
one. 

7.  Obtain  another  root  figure  in  the  manner  last  men- 
tioned, and  thus  continue  the  operations  as  far  as  neces- 
sary. 

Jfote  1. — In  multiplying  by  each  root  figure,  attention 
must  be  given  to  its  value.  Thus,  if  it  is  of  the  order  of 
tens,  the  multiplication  must  be  made  by  the  number  of 
tens  which  it  represents ;  and  so,  for  other  values.  Also, 
in  the  multiplications  and  additions,  regard  must  be  had  to 
the  signs  of  the  numbers. 

2.  The  signs  of  the  successive  sums  under  the  absolute 
number  and  last  co-efficient,  must  continue  the  same  through- 
out the  operation,  or  both  change  by  the  same  root  figure. 
14* 


162  equatio3n:s  in  general. 

If  the  operation  for  any  of  the  root  figures  causes  only  one 
sign  to  change,  another  value  must  be  taken.  This  will 
not  unfrequently  occur  with  regard  to  the  second  root 
figure,  but  it  will  seldom  be  the  case  for  the  others. 

3.  After  two  or  three  root  figures  have  been  obtamed, 
and  the  multiplications  and  additions  corresponding  to 
them  have  been  completed,  the  succeeding  parts  of  the 
operation  may  be  contracted  in  the  following  manner. 
Cut  off  the  right  hand  figure  of  the  sum,  in  the  column 
under  the  last  co-efficient;  the  two  right  hand  figures 
of  the  sum  in  the  preceding  column ;  the  three  right 
hand  figures  of  the  sum  in  the  column  preceding  that,  and 
so  on. 

If  either  of  the  figures  next  to  the  right  of  the  marks 
of  separation  is  5,  or  more  than  5,  add,  mentally,  a  unit 
to  the  first  figure  on  the  left  of  the  mark,  when  using 
it  in  the  succeeding  multiplication  and  addition.  Re- 
peat the  same  contraction  for  each  of  the  following  root 
figures. 

These  contractions  may  commence,  in  cubic  equations, 
after  the  second  or  third  decimal  figure  in  the  root  is  ob- 
tained ;  in  biquadratic  equations,  after  the  first  or  second 
decimal  figure ;  and  in  higher  equations,  after  the  first  de- 
cimal figure.  And  if  the  operation  is  closed  when  the  sum 
under  the  absolute  number  is  reduced  to  two  figures,  all 
the  figures  in  the  root  will  be  true.* 


EXAMPLES. 

1.    Given  3a?*--4a?3+ 2a:— 1000=0,  to  find  the  value 
ofx.  Ans.  4.342447603. 


•This  rule,  improved  from  Young's  Algebra,  was  communicated  by 
my  friend  John  Gummere,  of  Burlington. 


EQUATIONS   IN    GENERAL. 


163 


0 
12 

12 
12 

—4 
48 

44 
96 

140 
144 

284 
14.67 

+  2 
176 

178  ■ 
560 

738 
89.601 

—1000(4.342447603 
712 

—288 
248.2803 

24 
12. 

—39.7197 
37.39678208 

36 
12 

827.601 
94.083 

—2.32291792 
1.89781666 

48.9 
.9 

49.8 
.9 

50.7 
.9 

51.6 
.12 

51.72 
12 

51.84 
12 


298.67 
14.94 

313.61 
15.21 

328.82 
2.0688 


921.684 
13.235552 

934.919552 
13.318496 

948.23804,8 
.67028 


330.8888    948.90833 
2.0736  .67048 


332.9624    949.5788,1 
2.0784  .1340 


335.04,08  949.7128 
10  1340 


51.96 
.12 


335.14       949.846,8 
10  12 


5,2.08 


335.24 
10 


949.859 
12 


—.42510126 

37988512 

—4521614 
3799436 


—722178 
664909 


-57269 
56994 

—275 
285 


3,35.34    94,9.8,7,1 
2.  Given  a?5+2a7^+3a?3  + 4372+ 5a?— 54321— 0,  to  find  the 
value  of  07.  Ans.  8.41445475. 


164 

1     2 

8 

3 

80 

83 
144 

227 
208 

435 

272 

707 
16.96 

EQUATIC 
4 

664 

668 
1816 

2484 
3480 

5964 

289.584 

)NS   IN    GENERj 

5 
3344 

5349 

19872 

25221 
2501.4336 

-54321(8.41445475 
42792 

10 

8 

-11529 
11088.97344 

18 
8 

—440.02656 
304.11052 

26 

8 

27722.4336 
2620.0064 

—135.91604 
122.02904 

34 

8 

6253.584  30342.440,0 
296.432        68.612 

—13.88700 
12.21504 

42.4  723.96 
.4    17.12 

6550.016  30411.052 
303.344        68.690 

—1.67196 
1.52700 

42.8  741.08 
.4    17.28 

6853.3,60  30479.74,2 

7.8            27.52 

—14496 
12216 

43.2  758.36 
.4    17.44 

6861.2 

7.8 

6869.0 

7.8 

30507.26 

27.52 

—2280 
2135 

43.6 
.4 

77,5.80 
( 

30534.7,8 

2.8 

—145 
153 

44.0 

5,87,6.8 

30537.6 

2.8 

30,5,4,0.4 

3.  Given  a?^  +  9a?H4a?=80,  to  find  x. 

Result,  07=2.4721359. 

4.  Given  x^-\-x'^-\-x—^0^  required  the  value  of  a?. 

Ans.  07=4.1028323. 

5.  Given  a^^-f  lOa?^-]- 5^7=:  2600,  required  the  value  of  x, 

Ans.  11.00679934. 

6.  Given    2a?* +1607^+40073+3007= 4500,    required  the 
value  of  07.  Ans.  07=5.03770809. 


EQUATIONS  IN  GENERAL.  165 

134.  To  show  the  rationale  of  the  process  directed  in  the 
last  article,  I  begin  with  the  cubic  equation 

ax'^-{-bx'^  +  cx  +  d=0. 

Let  the  first  figure  in  the  root  be  indicated  by  r,  regard 
being  paid  to  its  local  value,  and  the  remaining  part  of  the 
root  by  y;  then  x=y  -^r. 

Hence,  ax^'=ay'^-\-3a7'y'^-{-3ar~y-}-ar^ ) 

hx'^  =  by^  +  2bry  +  br^    I  _  ^ 

cx=  cy-\-cr 

d^  d 

Collecting  the  co-efficients  of  like  powers  of  y, 
ay^+b'y'^-{-c'y-\-d'-=0. 

It  is  obvious  that  d'=\(ar-]-b)r-{-c\r'\'d; 

dz={ar'\-b)r-\-{^ar-\-h)r-\-c;  b' z=ar -\- ar '\-ar -{-b . 

But  these  are  the  quantities  found  by  the  fourth  precept. 

Again,  since  r^y,  c'y-\-d'  approximates  to  0,  or  y= 

— —  nearly ;  but  this  is  the  mode  prescribed  in  the  fifth 

precept  for  finding  the  next  figure  of  the  root. 

Denoting  the  number  obtained  by  the  last  operation  by 
5,  and  the  remaining  part  of  y  by  z,  so  that  y=^z-\-s^  we 
shall  obtain  a  new  equation  az^-{'b"z^  +  c"z  +  d"=^0,  in 
which  d'z=\(as-{-b')s  +  c'ls+d';  c"=(as+b')s-{-(2as+b')s 
-|-c';  b"=as-\-as+as+b';  whence  an  approximate  value  of 
2:,  or  a  new  figure  of  the  root,  is  manifestly  deducible  from 
this  new  equation,  as  before. 

Assuming  now  the  general  equation 

ax''-\-bx''~^ mx^-\-nx+p=^0. 


^==0 


166  INDETERMINATE  PROBLEMS. 

And  denoting  as  before,  the  first  figure  of  ^the  root  by  r, 
and  making  x=y-\-rj  we  have 

ax''=^ay''-{-nary^~^ etc.         nar^~^y'\-ar^ 

bx''-^=  by""-^,,        etc.  n — l.^r^-^y  +  Z^r"-^ 

etc.  ...  etc. 

mx'^=  2mry-^mr^ 

nx—  ny-^nr 

P=  P) 

Or  ay''  +  bY~^+"  etc.  n'y+p'=Oy  in  which  the  quantities 
^'5  ^'>  ?'•>  ^re  composed  of  the  co-efiicients  a,  b,  etc.  and  the 
powers  of  r  combined,  as  directed  in  the  fourth  precept. 

P' 
And  here  as  before  y= — =-j  nearly. 

Hence  it  is  obvious,  that  the  successive  figures  of  the 
root  may  be  obtained  by  the  same  kind  of  process ;  what- 
ever may  be  the  index  of  the  highest  power  of  the  un- 
known quantity. 


INDETERMINATE  PEOBLEMS. 

135.  When  a  problem  is  given,  in  which  the  number 
of  unknown  quantities  employed,  is  greater  than  the  num- 
ber of  independent  equations  furnished  by  the  conditions 
of  the  problem  5  one,  at  least,  of  those  quantities,  may  be 
assumed  at  pleasure.  Such  problems,  therefore,  generally 
admit  of  an  indefinite  number  of  answers.  There  are, 
however,  certain  conditions  sometimes  annexed,  by  which 
the  number  of  answers  is  partially  limited.  For  example, 
the  answers  are  required  to  be  whole  positive  numbers,  or 
they  are  required  to  be  square  or  cube  numbers.  Such 
problems  are  termed  indeterminate,  or  unlimited,  though, 
in  some  instances,  each  unknown  quantity  admits  of  but 
one  value.     If  simple  powers  only  of  the  unknown  quan- 


INDETERMINATE  PROBLEMS.  167 

titles  are  included  in  the  equations,  the  problem  is  said  to 
be  of  the  first  decree. 

Case  I. 

To  find  the  values  of  x  and  y,  in  whole  positive  numbers,, 
from  the  equation  ax=by-{-c;  a,  b,  c,  being  given  num- 
bers, positive  or  negative,* 

by-\-c 
Here  a?=-^ — ,   and  as  x  is  to  be  a  whole   number, 

by  -\-c 

must  also  be  a  whole  number. 

a 

Now,  if  this  quantity  be  multiplied  by  a  whole  number, 

the  product  must,  evidently,  be  a  whole  number;  also  the 

sum  or  difference  of  this  quantity,  or  either  of  its  multiples, 

and  any  whole  number,  must  necessarily  be  a  whole  num- 

bv  -\-c 
ber.     Let,  therefore,  — —  be  thus  chano-ed  till  we  obtain 

V  -4~  c 

=2^A,f   which  put  =J9;  then  y=ap — c',  a  quantity 

that  must  be  a  whole  number,  because  n,  p,  and  c'  are 

whole  numbers.     And  this  value  being  substituted  for  y  in 

by-\'C 
the  equation  x= ,  the  value  of  a?  will  be  obtained  in 

terms  of  j?,  and  given  numbers. 

Assuming  then  j9=0,  1,  2,  3,  etc.  successively,  (omitting 
such  numbers  as  make  x  or  y  negative,)  the  various  nu- 
merical values  of  x  and  y  become  known. 

The  number  of  answers  will  be  limited  when  the  signs 
of  p  in  the  values  of  x  and  y  are  unlike :  but  unlimited 
when  p  has  the  same  sign  in  both. 

=*  If  ff  and  b  have  a  common  divisor,  it  must  also  be  a  divisor  of  c, 
or  the  problem  is  impossible. 

fl'his  expression  is  used  to  designate  any  trholc  vumlcr. 


168  INDETERMINAAE  PROBLEMS. 


EXAMPLES. 


1.  Given  l9a?=14?/4-15,  to  find  the  values  of  x  and  y 
in  whole  positive  numbers. 

14y+15        ,         Uy  +  lb     , 

56?/+60     ^       ^     18^  +  3 

_,       19y        ^         19y     18y-f  3     y— 3    •   , 
But  --|=z.A.  ...  ^--A^^y—^^h^p 


...2/=19;?  +  3,  and  a?=— ^-i^-j^ — l—^=Up  +  3. 

If  now,  we  assume  p  successively  =0,  1,  2,  3,  etc. 

2/=  3,  22,  41,  60,  etc. 

a?=3,  17,  31,45,  etc. 
Here  the  number  of  answers  is  evidently  unlimited. 

2.  Given  lla?+l'73/=987,  to  find  a?  and  y  in  whole  posi- 
tive numbers. 

987— 17y     ^^  8— 6v        ,         8— 6v 

x= jj-^=-89— y  +  -^^=t^A.  .-.  — ^^=u'^, 

8—61/     ^     16— 12i/  5— y  V— 5 

y — ^     7 

Whence  y  =  1  Ip  +  5, 

987— (11;)+5)X17     ^^     ^^ 
And  a?:=- i—Y^-^^ =82— 17p. 


INDETERMINATE  PROBLEMS.  169 

Assuming  ^=0,  1,  2,  3,  4, 

2^=5,  16,  27,  38,49. 
a:=82,  65,  48,  31,  14 

Which  are  all  the  possible  values   in   whole   positive 
numbers. 

3.  Given  7a: +  93^= 2342,  to  find  the  number  of  values 
of  X  and  y  in  whole  positive  numbers. 

2342— 9  V     ^^,            2y— 4        ,        2y— 4 
x= = — ^=334 — y ^ — =wL  r.~^ — =wh. 

%-i..._8y-i6_    ,v-2_,      y-2 


■X4 


=y—2+^-Y-=wL  .'.^=wk=p. 


7 
.•.3/=7p+2,  and  a?=332~9i?. 


From  the  first  of  these  expressions  we  perceive  that 
the  least  value  of  p  is  0,  and  from  the  second,  that  the 
greatest  value  of  p  is  36.  Hence  the  required  number 
is  37. 

4.  Given  5a? +  73/ +92:  =337,  to  find  the  number  of  val- 
ues of  X,  y,  and  z^  in  whole  positive  numbers. 

3^7— 7y— 9z     ^^              ,  2— 2y— 4^       , 
x=:^ -^ =67 — y — z+ 1 z=zwh. 

^  2— 2y— 42r_ 
*  *         5 

.    ,  2— 2v— 42?     ^     4— 4v— 82: 
And g X  2= 1 =wk. 

52/— 5  +  102?        , 

But  -^ — ~ =wk. 

5 

52/-_5  +  102?     4— 4v— 82?     V— l  +  22r 

r,y=^p — 22? +  1,  x=^66 — 7/? +  2-. 
15 


170  INDETERMINATE  PROBLEMS. 

Assuming  now  z  successively  equal  to  1,  2,  3,  etc.  we 
shall  have  the  corresponding  values  of  x  and  ?/,  the  greatest 
and  least  values  of  jp,  and  the  number  of  answers  as  fol- 
lows: 


z 

X 

y 

P 

P 

Gr. 

Least. 

No.  of  Ans. 

1 

Ql—lp 

5p—l 

9 

1 

9 

2 

6S—7p 

6p—S 

9 

1 

9 

3 

69— Ip 

bp — 5 

9 

2 

8 

4 

10— Ip 

bp—1 

9 

2 

8- 

5 

71— Ip 

5;?— 9 

10 

2 

9 

6 

12— Ip 

5jo— 11 

10 

3 

8 

7 

13— Ip 

5p— 13 

10 

3 

8 

8 

1^—lp 

5j9— 15 

10 

4 

7 

9 

Ib—lp 

5/7—17 

10 

4 

7 

10 

16— Ip 

5;?-~19 

10 

4 

7 

11 

11— Ip 

5;?— 21 

10 

5 

6 

etc. 

etc. 

etc. 

etc. 

etc. 

etc. 

Here  we  may  observe,  there  are  seven  successive  values 
of^r,  (beginning  with  the  5th,)  which  produce  no  change 
in  the  greatest  value  of  p;  and  there  are  alternately  two 
and  three  equal  least  values  of  p;  ajid  this  order  will  evi- 
dently continue  as  long  as  successive  values  of  z  are  as- 
sumed. 

The  last  three  columns  of  the  above  table  may,  there- 
fore, be  continued  without  the  former  ones. 

Gr.  val.  oi  p,  9,  9,  9,  9,  10, 10,  10, 10, 10, 10,  10,  11,  11, 11, 
Least  do.         11222334445566 
No.  of  Ans.     a  9889     8     8777676     6 

Gr.  val.  of^,  11  11  11  11  12  12  12  12  12  12  12  13  13  13 

Least  do.  6    7    7    8    8    8    9    9  10  10  10  11  11  12 

No.  of  Ans.       65545544333332 

Gr.  val.  of  p,  13  13  13  13  14  14  14  14 
Least,  12  12  13  13  14  14  14  15 

No.  of  Ans.       2    2     111110 

The  number  of  answers,  or  sum  of  the  numbers  in  the 
last  column,  is  169. 


INDETERMINATE  PROBLEMS  171 

Otherwise — 

irom  the  equation  z— ^ ,  since  neither  a?, 

nor  3/  can  be  less  than  1,  it  is  obvious  that  2:  cannot  exceed 
36.  Then  in  the  equation  07=66 — Tp+ar,  make  2:=  36 
and  x=^  102 — 7^  ;  where  the  greatest  possible  value  of 
j3=14.  Put  p=14f  in  the  equation  y=op — 22r+l,  and 
2^=71— -22r.  From  which  we  perceive  that  the  greatest 
value  of  2;=  35.  Assuming  then  z=  1,  2,  3,  etc.  to  35,  and 
taking  all  the  values  of  p,  from  14  to  the  lowest,  which 
will  give  a  positive  value  to  y;  and  combining  the  results 
five  by  five,  we  shall  have  a  series  in  arithmetical  progres- 
sion, 67 -1-57+  .  .  .  7=259,  which  is  the  number  of  whole 
positive  values  of  y.  But  some  of  the  corresponding  values 
of  X  will  be  0  or  negative. 

From  the  equation  x=66 — Ip+z,  we  find  the  first  four 
values  of  z,  1,  2,  3,  4,  render  a?=0,  or  negative  if  p^9. 
Hence,  we  have  20  inadmissible  values  of  x.  For  the 
next  7  values  of  z^  we  have  x=0^  or  negative  if  p^lO ; 
which  give  28  inadmissible  values.  Proceeding  until 
z=35,  we  have  the  series  20 +  28-}- 21+  .  .7=90.  Hence 
259 — 90=169,  the  number  required. 

4.  Given  14a?=5?/+19,  to  find  the  least  possible  values 
of  X  and  t/,  in  whole  positive  numbers. 

Ans.  x=6^  y=l3. 

5.  Given  11a? +  5^=254,  to  find  all  the  values  of  x  and 
y,  in  whole  positive  numbers. 

Ans.  a?=19,  14,  9,  4. 
y=9,20,31,42. 

6.  Given  9a?+133/=2000,  required  the  number  of  values 
of  X  and  y.  Ans.  17. 

7.  Required  to  divide  100  into  two  parts,  so  that  one  of 
them  may  be  divisible  by  7,  and  the  other  by  11 1 

The  parts  are  56  and  44. 


172  INDETERMINATE  PROBLEMS. 

8.  Given  17a?4-193^+2l2r=400,  required  the  number  of 
values  of  ^,  y,  and  z,  Ans.  10. 

9.  Required  to  pay  1000  dollars,  in  French  crowns,  and 
five  franc  pieces,  so  that  the  number  of  coins  used  shall  be 
the  least  possible  \  what  number  of  each  kind  will  be  ne- 
cessary, and  how  many  ways  can  that  sum  be  paid  in  those 
coins,  the  French  crown  being  1 10  cents,  and  the  five  franc 
piece  93  cents  1 

Ans.  833  crowns,  90  five  franc  pieces,  and  9  different 
ways. 

10.  How  many  gallons  of  liquor  at  12  cents,  15  cents, 
and  18  cents  per  gallon,  may  be  mixed,  to  compose  300 
gallons  at  17  cents  per  gallon  1* 

C  at  12  cents,  1,  2,  3,  etc.  to  49. 
Ans.  \  at  15  cents,  98,  96,  94,  etc.  to  2. 
I  at  18  cents,  201,  202,  etc.  to  249. 

11.  In  how  many  ways  can  £1053  sterling  be  paid 
without  using  any  coins  besides  guineas  and  moidores ;  the 
guinea  being  2  J 5.  sterling,  and  the  moidore  271 

Ans.  112  ways. 

12.  A  foreigner  having  a  bill  of  $1  75  to  pay  at  a  hotel, 
offers  napoleons  in  payment,  the  landlord  agrees  to  receive 
them  on  condition  that  Turkish  sequins  shall  be  taken  as 
change  ;  how  many  pieces  must  be  used,  a  napoleon  being 
worth  $7  25,  and  a  sequin  rated  at  $2  10  \ 

Ans.  35  napoleons,  and  120  sequins. 

13.  Given  7a7  4-92/-|-232r=9999 ;  of  how  many  values 
will  0?,  2/,  and  z  admit  % 

Ans.  34365. 


,     *  When  more  equations  than  one  are  given,   one,  at  least,  of  the 
unknown  quantities  may  be  eliminated,  and  the  equations  reduced  to 


indeterminate  problems.         1v3 

Case  2. 

136.  To  find  a  whole  number^  which ^  being  divided  by  any 
given  numbers^  shall  leave  given  remainders. 

Denoting  the  required  number  by  a?,  the  given  divisors 
by  a,  Z>,  c,  etc.  and  the  remainders  by  /,  g,  A,  etc.  we 

shall   have  ,  — j — , ,  etc.  severally  equal  to  whole 

numbers. 

/p f 

Put  — -  =p.  find  the  value  of  x.  and  substitute  it  for  x 
a       ^' 

in  the  second  fraction ;  reduce  this  new  fraction,  as  in  the 
former  case,  so  that  the  co-efficient  of  p  may  be  a  unit,  and 
put  the  fraction  thus  obtained  =q;  find  the  value  of  p, 
and  thence  of  x,  in  terms  of  q  and  given  numbers.  Sub- 
stitute the  value  of  x  in  the  third  equation,  and  so  proceed 
The  last  value  of  x  will  be  the  number  sought. 

EXAMPLES. 

1.  Required  a  number,  which,  being  divided  by  7,  9, 
10,  and  11,  shall  leave  the  remainders  5,  4,  7,  and  9,  re- 
spectively. 

Let  x=i  the  number  sought. 

_,       X — 5  X — 4  X — 7  X — 9  ,    .  , 

Then  -y-,  -^,  -jg"'  TF'  ^^^  numbers. 

—--=:wh—p,       .-.O^rrrTp  +  S. 

7^+5-4    7^+1        , 
Whence,  • ^ =  -f^-^—.-=wh. 


7p-fl     ,     28;?+4     ^       P-f4^        . 


15* 


174?  INDETERMINATE  PEOBLEMS. 

79+4 


9 


-=wk=q,  nnd p~9q — 4.     r,x=63q — 23. 


And  consequently. 

^*  =wh-=r,     .-.^^lOr,  anda?=630r— 23. 

630r— 32     ,^       ^     3r+l        ,  3r  +  l 

.-. ==51r—3+-jY-=wL       .-.  —j—=zwh. 

^    ,3r+l     ^      12r  +  4  r— 7 

And-jPx4=--^j-=r+l+--^==2^A. 

r— 7 
.•.— — -='i^A=5.     .-.r— 115  +  7. 

And  a:=  69305  4- 4387=  (if  5=0)4387. 
Assuming  5=  1,  2,  etc.  other  values  would  arise. 

2.  Eequired  a  number,  which,  being  divided  by  2,  3,  4, 
6,  shall  leave  no  remainder;  but  being  divided  by  7,  the 
remainder  shall  be  6  1 

The  least  common  multiple  of  2,  3,  4,  6,t  is  12. 


•  If  a  fractional  expression  -  ri=i^?A,  and   we   divide   the   terms  into 

0 

iheii  primes,  it  is  plain  that  all  the   prime  numbers   contained   in 
the  denominator,  must  be  also  contained  in  the   numerator ;    thus, 

^-  z=:^'^z=wh,  where  the  primes  2  and  5  not  being  contained  in  the  3> 
10     2.5 

must  be  contained  in  <?,  and  therefore,  — =<»/;^. 

10 

fSee  Aritheraeticai  Expositor,  Part  1,  page  90. 


INDETERMINATE  PROBLEMS.  175 

Let,  therefore,  12x—  the  number  sought. 

12a?— 6        ,         12a?— 6           36a:— 18 
.-. — y-==w;A.  .-. — ,^— x3= ==:5a?— 2-i- 

=iwh;  and  -——z=wh=p,     .•.a?=7p  +  4. 

And  12a7=:84p  +  48=48,  132,  216,  etc. 

3.  Required  the  least  whole  number,  which,  being  di- 
vided by  3,  5,  7,  and  8,  shall  leave  the  remainders  1,  3,  5, 
and  0,  respectively  %  Ans.  208. 

4.  Required  the  least  whole  number,  which,  being  di- 
vided by  9,  10,  11,  and  12,  shall  leave  the  remainders  4, 
5,  6,  and  7,  respectively  1  Ans.  1975. 

5.  What  is  the  least  whole  number,  which,  being  di- 
vided by  each  of  the  nine  digits,  shall  leave  no  remainder ; 
but  divided  by  17,  the  remainder  shall  be  13  1 

Ans.  40320. 

6.  In  what  year  of  the  Christian  era  was  the  solar  cycle 
15,  the  lunar  cycle  3,  and  the  Roman  indiction  141  * 

Ans.  1826. 

7.  Required  the  least  whole  number,  which,  being  divided 
by  2,  3,  4,  and  5,  shall  leave  a  remainder  of  1  ,•  but  divided 
by  7,  there  shall  nothing  remain  1  Ans.  301. 

*  The  solar  cycle  is  a  period  of  28  years,  at  the  end  of  which,  in 
case  a  centurial  year  has  not  intervened,  the  days  of  the  week  always 
return  to  the  same  days  of  the  month.  To  the  year  of  the  Christian 
era  add  9,  and  divide  by  28,  the  remainder  will  be  the  number  of  the 
cycle.  ^    ^ 

The  lunar  cycle  is  a  period  of  19  years,  at  the  expiration  of  which 
the  new  and  full  moons  return  nearly  to  the  same  time  of  the  year. 
To  the  year  of  the  Christian  era  add  1,  and  divide  by  19,  the  remain- 
der is  the  lunaf-  cycle. 

The  Roman  indiction  is  not  an  astronomic  period,  it  consists  of  15 
years.  To  a  given  year  add  3,  and  divide  by  15,  the  remainder  is  the 
number  of  the  indiction. 

If,  in  either  of  these  cases,  no  remainder  occurs,  the  divisor  must 
be  taken  as  the  number  of  the  cycle. 


17G  MISCELLANEOUS    EXAMPLES. 


MISCELLANEOUS  EXAMPLES. 

137. — 1.  A  bankrupt  owes  A.  twice  as  much  as  he  owes 
B.,  and  he  owes  to  C.  as  much  as  to  A.  and  B.  together; 
the  sum  to  be  divided  is  600  dollars  ;  how  much  must  each 
receive'?  Ans.  A.  200,  B.  100,  C.  300. 

2.  A.  can  perform  a  piece  of  work  in  7  days,  and  B.  in 
9  days ,  in  what  time  would  they  jointly  effect  it '? 

Ans.  3  51  days. 

3.  A  laborer  being  employed  on  condition,  that  for  every 
day  he  worked  he  should  receive  50  cents,  and  for  every 
day  he  was  idle,  he  should  forfeit  20  cents,  finds  at  the  qnd 
of  500  days  only  89  dollars  due ;  how  many  days  did  he 
work'?  Ans.  270. 

4.  There  are  two  numbers  in  the  ratio  of  4  to  5,  and  the 
sum  of  their  squares  is  1476  ;  what  are  the  numbers '? 

Ans.  24  and  30. 

5.  A  Greek  epitaph,  designed  for  the  tomb  of  Diophan- 
tus,  is  said  to  have  stated  that  he  passed  one-sixth  of  his 
life  in  childhood ;  one-twelfth  in  adolescence ;  that  after 
one-seventh  and  five  years  more  had  been  passed  in  a 
married  state,  he  had  a  son  who  lived  to  half  his  own  age, 
and  whom  he  survived  four  years ;  what  then  was  the  age 
of  Diophantus  '?  Ans.  84  years. 

6.  A  person's  age  in  years  is  a  number,  consisting  of  two 
digits ;  ^  of  this  number  is  a  mean  proportional  between 
these  digits ;  and  two  years  hence  his  age  will  be  a  third 
proportional  to  those  digits,  beginning  with  the  tens ;  what 
IS  the  age  '?  Ans.  14  yearsi 

7.  Givena7  +  y  +  2r=9,  xy -^xz -\-yz=26,  xyz =24^,  to  ^nd 
the  values  of  a?,  y,  and  z*  Result,  a?=2,  3,  or  4. 

*  If  we  assume  the  equation  v^ — 9^3^26^ — 24=0,  then  art.  126,  the 
roots  of  this  equation  will  be  the  values  required ;  and  those  roots  are 
easily  found  by  art.  132. 


MISCELLANEOUS  EXAMPLES.  177 

8.  By  selling  a  piece  of  muslin  at  a  certain  price  per 
yard,  I  gained  the  prime  cost  of  9  yards,  which  was  just  as 
much  per  cent,  as  the  number  of  yards  in  the  piece ;  what 
was  that  number '?  Ans.  30  yards. 

9.  There  are  four  numbers  in  continued  proportion,  the 
sum  of  the  first  and  last  1728,  and  the  sum  of  the  other  two 
1152  5  what  are  the  numbers  1 

Ans.    192,  384,  768,  and  1536. 

10.  The  hypothenuse  of  a  right  angled  triangle  is  3?^% 
and  the  other  sides  a?^^,  and  x^;  what  is  the  areal 

Ans.  W(2+V5). 

}.l.  There  is  a  number,  consisting  of  three  digits  in 
arithmetical  progression,  which  number  being  .divided  by 
the  sum  of  the  digits,  the  quotient  will  be  48,  but  if  from 
the  number  198  be  subtracted,  the  remainder  will  be  ex- 
pressed by  the  same  digits  in  an  inverted  order.  Quere 
the  number  1  Ans.  432. 

12.  The  sum  of  the  squares  of  two  numbers  being  mul- 
tiplied by  the  quotient  arising  from  the  division  of  the  less 
by  the  greater,  produces  83.2,  and  the  difference  of  the 
squares  multiplied  by  the  quotient  of  the  greater  divided 
by  the  less,  produces  1920  ;  what  are  the  numbers  1 

Ans.  20  and  4. 

13.  A  ball  falling  from"  the  top  of  a  tower,  is  observed 
to  descend  one-fourth  of  the  distance  in  the  last  second  of 
the  time ;  required  the  height  of  the  tower,  heavy  bodies 
being  known  to  fall  16  J^  feet  during  the  first  second,  and 
to  describe  spaces,  which,  reckoned  from  the  beginning  of 
the  fall,  are  as  the  squares  of  the  times  "l 

Ans.  (28+16^3)16^2  feet. 

14.  What  are  the  values  of  x  and  z,  from  the  equations 
x'z+xz^=b4^6bQ0,  x^+z^  =10869921 

Ans.  a?=32,  2r=14. 

15.  GiYen  x+y+z+v=:56,  x''+y^'^z^+v^=910,  xv-\- 
22/2— 2^2=6,  and  z=2y^  to  find  the  values  of  a?,  y,  z,  and  v 

Result,  x=S,  y=:9,  2:=  18, 2;=21. 


178  MISCELLANEOUS  EXAMPLES. 

16.  Required  to  find  a  number,  which,  being  any  way 
divided  into  two  unequal  parts,  the  greater  part  added  to 
the  square  of  the  less,  shall  be  equal  to  the  less  part  added 
to  the  square  of  the  greater.  Ans.  1. 

17.  Given  ccy=n5x+300y,  and  2/3— 0?^=  90000,  to  find 
X  and  y  by  a  quadratic*  Result,  a?=400,  y=500. 

18.  Given  (a?-^+l).(a?^+l).(a?+ l)  =  30a?^  to  find  the 
value  of  X  by  a  quadratic  equation.! 

Result,  a:=i(3±v/5). 

19.  The  sum  of  three  numbers  in  harmonical  proportion 
is  26,  and  their  continued  product  576  ;  required  the  num- 
bers by  a  quadratic  equation.  Ans.  12,  8,  and  6j^ 

*  This  is  readily  solved  by  quadratics,  by  finding  x  in  terms  of  y 
from  the  first  equation,  and  substituting  the  value  in  the  second. 

fThis  question  requires  expedients  not  readily  found. 
By  multiphcation  and  transposition. 

x^-j^x^+x* — 280^3  4- a?2+a?+ 1  =  0,  whence 

oo  1  1  1  ^ 

But  2/=a?-f-,  whence  y^-fy2 — 2y — 30=0 

X 

1       ,  '         7      790     79  10      . 

Assume   z=y+p  wherefore  z^^-z=-^=— ,— 

'7         '79  10        ,^^.       (lO^V 
And  z* — ^z^=^  "q"'"^^-     Adding  ^-^ —  to  each  member 

79         10'         79  10 

^*+T^^=-(T")^+y-3" 

Hence  2r=-jr-,  and  y=3.     Whence  x^ — 30?= — 1. 

3zfcs/5 

and  x  =  — tr — 


MISCELLANEOUS  EXAMPLES.  179 

20.  The  sum  of  two  numbers  is  152,  and  the  cube  root 
of  the  square  of  their  difference  multiplied  by  the  square 
root  of  the  cube  of  the  same  difference,  produces  8192; 
what  are  the  numbers  1  Ans.  44  and  108. 

21.  The  sum  of  two  numbers,  added  to  the  sum  of  their 
squares,  is  120,  and  the  product  of  the  same  numbers  45 ; 
what  are  they  %  Ans.  9  and  5. 

22.  Given  x^+xy^=z4<64}0y^  and  x'^y — 2/3=  537.6a?;  what 
are  the  numerical  values  of  x  and  y  ? 

Ans.  X — 40,  y=16, 

23.  GiYen  x-\-y=:z,  z^ — x^+y^=4}10,  z^ — x^ — y'^—^Q'^^ 
to  find  0?,  y,  and  z.  Result,  x=  12,  y=  1,  z— 13. 

(      .3 

24.  Given  <  x-\-y'  +a?+2/=30. 

(  X — y=  1.     Required  the  values  of  x  and  y, 
Ans.  a?=2,  y=l. 

25.  Given  x^ — 2a:3-f  a7=«,  to  find  a?  by  a  quadratic  equa- 
tion. Result,  a:=  4  +  ^/  (I  +  s/~^i), 

26.  Given  c(f^+x[y-\-z)=a^  y^'{-y(x+z)=b,  z^+z{x-{-y) 
=  c,  to  find  X,  y,  and  z. 


Results,  ^ 


b 

c 
'  y/(a+b+c) 


27.  Given  2a:3 — x* — x'^'\-2x^y — y^=^2xy—  s/x^y — s/x^y, 
to  find  X  and  y.  Result,  x=5,  y—20, 

28.  Required  the   roots   of  the   equation   4fOC^-\-Sx^ — 
89a?2+28a?+49=0,  by  quadratics  only. 

7  —13+  V 1 13  — 13— v/  113 
Ans.  1,  ^, ^  , J 


180  MISCELLANEOUS  EXAMPLES. 

29.  There  are  three  numbers  in  geometrical  progression, 
whose  continued  product  is  4096,  and  the  sum  of  the  ex- 
tremes is  68  j  what  are  the  numbers  % 

Ans.  4",  16,  and  64. 

30.  There  are  two  numbers  expressed  by  the  same  two 
digits,  and  the  difference  of  their  squares  is  1485 ;  quere 
the  numbers  ]  Ans.  14  and  41. 

Or  78  and  87. 

31.  What  two  numbers  are  those  whose  product,  differ- 
ence of  their  squares,  and  quotient  of  their  cubes,  are  all 
equan  Ans.  3 +  ^^5^  and  i  +  W5. 

32.  The  product  of  two  numbers  is  10,  and  the  product 
of  their  sum  by  the  sum  of  their  squares  is  203 ,  required 
the  numbers  found  by  a  quadratic  equation. 

Ans.  5  and  2, 

33.  There  are  three  numbers  in  geometrical  progression, 
the  difference  of  whose  differences  is  6,  and  the  sum  of  the 
numbers  42,  what  are  the  numbers] 

Ans.  24,  12,  and  6. 

34.  There  are  three  numbers  in  harmonical  proportions, 
the  difference  of  whose  differences  is  1,  and  the  product  of 
the  extremes  18  -,  what  are  the  numbers  1 

Ans.  3,  4,  and  6. 

35.  Required  three  equi-different  numbers,  such  that  if 
the  first  be  increased  by  1,  the  second  by  2,  and  the  third 
by  the  first,  the  sums  may  constitute  an  harmonical  pro- 
gression ;  but  if  3  be  added  to  the  second,  the  sum  may  be 
a  mean  proportional  between  the  sum  of  the  numbers  and 
the  first  diminished  by  k*  Ans.  5,  6,  and  7. 

36.  Required  two  such  squares,  that  their  difference 
shall  be  to  the  square  root  of  the  less  as  3  to  7,  and  the 
square  roots  of  the  numbers  to  each  other  as  5  to  2. 

Ans.  (loy,  and  (^%y. 

37.  Given  the  sum  of  the  cubes  of  two  numbers  =35, 
and  the  sum  of  their  9th  powers  =20195;  what  are  the 
numbers  1  Ans.  3  and  2 


MISCELLANEOUS  EXAMPLES.  181 

38.  Given  9  -.  +  36^=85,  -^+-^  =  —  ^16, 
required  the  values  of  x  and  y.  Ans.  a:==3^,  y~2. 

39.  What  two  numbers  are  those  whose  difference  is  4, 
and  their  product  multiplied  by  the  sum  of  their  squares 
480  1  Ans.  6  and  2. 

40.  Giyen  x^y'^xi/=a,  xz^'s/xz=b,y^Zs/zy=c,  to  find 
the  values  of  x,  y,  and  z. 


41.  Required  the  values  of  a?,  y,  and  2r,  from  the  equa- 
tions yz=a,  xz=.h^  xy=c, 

.  be  ac  ah 

42.  There  are  three  numbers  in  harmonical  proportion, 
the  sum  of  the  first  and  third  is  54,  and  their  continued 
product  15552 ;  what  are  the  numbers'? 

Ans.  18,  24,  and  36. 

43.  In  how  many  different  ways  is  it  possible  to  pay 
JBIOOO,  without  using  any  other  coins  than  crowns,  guineas, 
and  moidoresj  a  crown  being  55.,  a  guinea  2l5.,  and  a 
moidore  275.  %  Ans.  70734. 

44.  Given  x''+y^-^z^=2QQ.^,  a?'2+y+;s2^  176.5, 

{a?-fy+2:)y=286.     Required  the  values  of  x 
2/,  and  z,  Ans.  07=10.5,  2/=  10,  2r=7.5. 

45.  Required  two  cube  numbers,  such  that  the  first  mul- 
tiplied into  the  product  of  their  roots,  shall  be  equal  to  the 
second ;  but  the  second  multiplied  into  the  product  of  their 
roots,  shall  be  equal  to  64  times  the  first. 

Ans.  8  and  64. 
16 


182  MISCELLANEOUS  EXABIPLES. 

46.  What  number  is  that  which  being  added  to,  and  sub- 
tracted from  36,  the  sum  of  the  cube  roots  shall  be  6  1 

Ans.  28. 

47.  Given  ^2+3/2— a?—i/=249740,  a??/  +  a?+?/=;:8516,  to 
determine  the  values  of  x  and  y. 

Result,  x=zb00,"y=:16. 

48.  Given  x'-j-y'-\-x''+y^=^23S63'2—2xY, 

x^+y^  +  z^=: '.y  (216100— x^—2f—z%) 
to  find  the  values  of  a?,  y,  and  z, 

Ans.  a:=22,  ^=2,  z=6, 

49.  The  sum  of  five  numbers  in  geometrical  progression 
is  242,  and  the  fourth  difference  is  32,  required  the  num- 
bers. Ans.  2,  6,  18,  54,  and  162. 

50.  Given  a?-f^/  +  2r+i;+w=12.15=a, 

X'\-y+z-\-vw=9.16  =  bj 
x-i-y-\-zvw=^6.6S=c, 
^         w -\- y  zvw =6. 09  =-d, 

xyzvw=A5—e,     Required  the  values  of 
a?,  y,  z,  V,  and  w, 

,  Ans..=^^'^(|=^i)  =  5,or.09. 

_  c—x±^  sf  \  {c—xy—^{<i—x)  \ 

y-  2 


a 


b — 0?—  2/=h  s/  \  (b — X — yf — 4(c — a? — y  \ 
; — X — y — 2r-_4=  V  \  (a — X — y — zy — 4(5 — x — y — z)  j 


2 
w=a — X — y — z — V. 

From  the  ambiguity  of  the  signs,  it  is  manifest  that  the 
number  of  numerical  values  of  the  unknown  quantities  go 
on  increasing  from  x  to  v. 


THE  END. 


ROBERT  E.  PETERSON, 
BOOKSELLER  AND  PUBLISHER, 

N.   W.   CORNER    OF    FIFTH   AND  ARCH  STREETS, 

PHILADELPHIA, 

Invites  attention  to  the  following  valuable  works, 
just  published. 

FAMILIAR  SCIENCE, 

OR 

THE  SCIENTIFIC  EXPIINATION  OF  COfflON  THINGS. 

EDITED  BY  R.  E.  PETERSON, 

MEMBER  OF  THE  ACADEMY  OF  NATURAL  SCIENCES,  PHILADELPHIA. 

12mo.  Sheep,    550  Pages.    Price  75  Cents.    A  deduction 
made  to  Teachers. 


The  attention  of  Teachers  is  invited  to  the  following  opinions  of 
the  work : 

From  Professor  W.  H.  Allen,  President  of  Girard  College  for  Orphans, 
Philadelphia. 

Girard  College,  May  6,  1851. 
Robert  E.  Peterson. — Dear  Sir — I  beg  leave  to  tender  my 
thanks  for  your  courtesy  in  sending  me  a  copy  of  "Familiar 
Science."  I  have  read  parts  of  each  division  of  the  work,  and 
have  been  pleased  with  the  precision  of  the  questions,  and  the  ac- 
curacy of  the  answers.  The  book  is  not  merely  a  volume  of /«• 
vniliar  knowledge^  but  a  volume  in  which  much  rare  and  profound 
knowledge  is  made  familiar  to  the  common  mind  and  applied  to 
common  things.  I  consider  the  book  a  valuable  contribution  to 
our  means  of  instruction  in  schools,  and  hope  to  see  it  generally 
introduced  and  used  by  teachers.  Fathers  of  families  also,  who  are 
now  frequently  puzzled  by  the  questions  of  the  young  philosophers 
of  their  households,  will  do  well  to  procure  a  copy  and  avoid  saying 
so  often,  "  I  do  not  know.''''  I  remain  truly  yours,  etc., 

Wm.  H.  Allen. 


2 

From  the  Rev.  Lyman  Coleman,  D.  D.,  Principal  of  the  Presbyterian  In- 
stitute, Philadelphia. 

"Familiar  Science"  embodies  a  vast  amount  of  facts  and  princi- 
ples, relating  to  the  several  branches  of  natural  science,  judiciously 
selected  and  arranged,  and  very  useful  to  awaken  inquiry  in  the 
young,  and  form  a  taste  for  such  studies.  For  this  purpose  it  is 
used  as  a  text  book  in  our  school. 

Lyman  Coleman,  Prin. 

PniLADELrHiA,  May  15,  185L 

From  the  Right  Rev.  Bishop  Potter. 

Broad  Street,  June  2,  1851. 
Dear  Sir — Absence  from  the  city  and  urgent  engagements  have 
prevented  me  from  acknowledging,  as  promptly  as  I  wished,  your 
note  of  last  month.  The  same  causes  have  prevented  ray  giving 
to  your  work,  entitled  "Familiar  Science,"  the  thorough  examina- 
tion to  which  its  apparent  merit  richly  entitles  it.  So  far  as  I  have 
been  able  to  look  through  it,  it  fully  justifies  your  selection  of  it 
and  the  laborious  revision  which  you  have  bestowed  upon  it.  It 
contains  a  vast  amount  of  useful  information  on  subjects  which 
force  themselves  upon  the  attention  of  both  old  and  young,  and  it 
is  likely  to  cultivate  in  those  who  read  it  habits  of  inquiry  and  re- 
flection. The  mechanical  execution  is  entitled  to  unqualified 
praise.  Very  respectfully  yours, 

Alonzo  Potter. 

Mr.  R.  E.  Peterson. — Dear  Sir— I  am  much  pleased  with  the 
volume  you  left  me  for  perusal,  entitled  "  Familiar  Science,  or  The 
Scientific  Explanation  of  Common  Things."  It  treats,  in  a  familiar 
and  interesting  manner,  of  a  large  variety  of  subjects,  which  all 
children  should  understand,  but  many  of  which  are  indifferently  un- 
derstood by  persons  of  mature  age.  The  book  cannot  fail  to  inter- 
est the  inquisitive  pupil,  and  become  a  valuable  auxiliary  in  the 
diffusion  o{ practical  science,  both  in  schools  and  families. 

Yours  respectfully, 

J.  Simmons, 
Principal  of  "  The  Locust  Street  Institute  for  Young  Ladies." 
Philadelphia,  May  17,  1851. 

FromN.  C.  Brooks,  A.  M.,  Principal  of  the  Baltimore  Female  College. 
Late  Principal  of  the  Baltimore  High. School. 

Baltimore,  May  28,  1851. 
Having  examined  the  American  edition  of  Dr.  Brewer's  Fa- 
miliar Science,  with  additions  by  R.  E.  Peterson,  I  take  pleasure 
in  commending  it  to  teachers  and  parents,  and  all  others  interested 
m  the  cause  of  education,  as  a  most  excellent  work.  It  is  worthy 
of  very  extensive  circulation.  I  shall  most  prooably  have  it  used 
as  a  text  book  in  our  Institution. 

N.  C.  Beooks 


From  the  Hon.  Joel  Jones,  Ex- president  of  Girard  College  for  Orphans 
Philadelphia  * 

1  take  great  pleasure  in  concurring  in  the  foregoing  recommenda- 
tion. JoEii  Jones. 

_  Having  carefully  exammed  the  book  entitled  "Familiar  Science," 
i  accord  to  it  my  decided  approbation. 

As  a  class  book,  in  the  hands  of  a  judicious  teacher,  it  cannot  fail 
to  strengthen  the  intellect  and  enlarge  the  useful  knowledge  of  all 
young  persons  that  may  study  it.  So  much  am  I  impressed  with 
its  value,  that  I  think  it  must  speedily  find  its  way  into  all  good 
schools. 

Edmund  Smith,  A.  M. 
"Franklin  Hall,"  Newton  University,  Baltimore, 

June  2,  1851. 


Burlington,  May  24,  1851. 
Robert  E.  Peterson,  Esq. — Dear  Sir — It  gives  me  great  plea- 
sure to  acknowledge  the  receipt  of  a  late  reprint  of  yours  entitled 
"Familiar  Science."  It  affords  another  of  the  many  examples, 
which  are  constantly  being  furnished  nowadays,  of  the  increased 
facilities  of  instruction.  Such  books  as  these  implant  in  the  youth- 
ful mind  a  desire  and  determination  to  inquire  more  extensively 
into  science,  and  thus  no  one  can  calculate  their  value,  for  we  are 
daily  called  upon  to  acknowledge  the  immense  advantages  which 
an  insight  into  chemical  action  brings  to  bear  upon  mechanical,  as 
well  as  agricultural  fields  of  labor.  I  can  well  imagine  the  plea- 
sure which  a  perusal  of  this  little  book  of  yours  will  afford  to  those, 
who  first  glean  from  its  pages  a  knowledge  of  the  subjects  on  which 
it  treats,  from  a  recollection  of  the  feelings  with  which  I  used  to 
read  a  much  more  unpretending  volume  of  the  same  nature,  now 
out  of  print,  entitled  "  Common  Things."  Assuring  you  that  it  will 
give  me  great  pleasure  to  recommend  your  book  as  an  elementary 
one  in  the  Institution  with  which  I  am  connected, 
I  have  the  honor  to  subscribe  myself 

Very  respectfully  yours,        George  H.  Doane, 
Instructor  in  Chemistry,  and  Lecturer  on  Physiology  at  Burlington  Col- 
lege. 


From  Miss  Phelps,  Philadelphia. 
Y"our  excellent  book,  "Familiar  Science,"  has  been  received, 
and,  after  having  carefully  examined  it,  I  am  prepared  to  concur 
entirely  with  the  numerous  testimonials  you  have  already  received 
of  its  merit  as  a  text  book  for  schools  or  families.  It  is  eminently  a 
book  for  the  times  and  for  the  people,  as  a  vast  amount  of  general 
information  on  useful  and  practical  subjects  may  be  acquired  with- 
out exploring  multiplied  abstruse  matter  for  learning,  compara- 
tively, a  few  facts.    I  predict  for  it  a  ready  and  rapid  sale,  and  con- 


gratulate  educators  in  having  so  able  an  auxiliary  in  the  business  (jt 
instruction.    Thank  you  for  the  presentation  of  a  volume,  and 
Am,  very  truly,  H.  M.  Phelps, 

Prin.  St.  Mary's  School. 


Philadelphia,  May  13, 1851. 
Mr.  Robert  E.  Peterson. — Dear  Sir — Many  thanks  to  you  for 
your  present  of  "  Familiar  Science ;"  I  have  given  it  a  careful  pe- 
rusal, and  believe  it  to  be  better  calculated  to  foster  in  the  mind  of 
3'^outh  a  spirit  of  philosophic  and  scientific  inquiry  than  any  work  1 
have  ever  seen.  The  questions  are  precise  and  the  answers  accu- 
rate ;  and  the  common  things  of  every  day  life  receive  a  profound 
and  luminous  explanation.  The  work  is  admirably  arranged,  and 
the  index  remarkably  complete.  It  should  be  in  all  the  schools  in 
the  country.  I  shall  introduce  it  into  mine  at  the  next  term. 
With  great  respect,  your  friend, 

D.  R.  ASHTON, 

Female  Institute,  Philadelphia. 

Weccacoe  Boy's  Grammar  School,  May  15,  1851. 

Sir — From  an  examination  of  the  book  entitled  "Familiar 
Science,"  which  you  were  pleased  to  send  me,  I  feel  convinced 
that  it  would  be  a  valuable  addition  to  the  list  of  text-books  now  in 
use  in  our  schools. 

The  importance  of  the  subjects,  and  the  familiar  manner  of  treat- 
ing of  them,  should  make  it  a  popular  family,  as  well  as  school 
book. 

John  Joyce,  Prin. 

Mr.  Robt.  E.  Peterson. 

May  16, 1851. 
Dear  Sir — The  pages  of  "Familiar  Science"  are  its  best  recom- 
mendation. The  common  phenomena  of  life  are  treated  of  in  a 
simple  and  intelligible  manner,  which  renders  it  both  pleasing  and 
instructive.  In  the  family  circle,  as  a  text  book,  it  will  form  the 
basis  of  an  hour's  interesting  conversation,  and  in  the  hands  of  the 
pupil  it  will  be  a  valuable  aid  in  the  acquisition  of  useful  know- 
ledge. Respectfully, 

Wm.  S.  Cleavenger, 
Prin.  Boys'  Grammar  School,  Locust  St. 
To  Robert  E.  Peterson,  Esq. 

Philadelphia,  May  17,  1851. 
Dear  Sir — I  am  highly  pleased  with  the  treatise  on  "  Familiar 
Science,  or  the  Scientific  Explanation  of  Common  Things,"  of 
which  you  sent  me  a  copy  for  examination.  I  have  examined  the 
work  and  find  it  contains,  within  a  comparatively  small  compass,  a 
great  amount  of  useful  information.    It  is  a  work  that  should  be  in 


the  hands  of  every  person,  both  young  and  old,  who  has  any  desire 
to  become  acquainted  with  the  common  phenomena  of  life. 
Respectfully  yours,  etc. 

Samuel  F.  Watson 
Prin.  of  Mount  Vernon  Boys'  Grammar  School. 
EloBERT  E.  Peterson,  Esq. 

Robert  E.  Peterson,  Esq. — ^Dear  Sir — I  have  been  much  gratified 
by  an  examination  of  your  book  entitled  "Familiar  Science,"  and 
believe  its  introduction  into  schools  and  families  would  do  more  in 
imparting  a  general  knowledge  of  scientific  principles  than  any  trea- 
tise ever  used. 

The  cause  of  every  day  phenomena,  such  as  evaporation,  con- 
densation, the  formation  of  cloudsj  rain,  dew,  etc.,  are  so  familiarly 
explained,  that  all  classes  of  persons  may  readily  comprehend 
them,  and  I  believe  the  book  has  only  to  be  known  to  be  appre- 
ciated by  teachers.  I  am  truly  yours, 

Wm.  Roberts, 
Prin.  of  Ringgold  Grammar  School. 

Philadelphia,  May  19,  1851. 

Philadelphia,  May  13, 1851. 
Robert  E.  Peterson,  Esq. — ^Dear  Sir — I  am  highly  pleased  with 
your  "Familiar  Science."  As  a  teacher,  I  am  constantly  called 
upon  to  answer  questions  upon  almost  every  conceivable  subject, 
and  children  wish,  and  should  have  correct  and  intelligent  answers. 
Your  two  thousand  questions  and  answers,  which  I  consider  most 
admirable,  will  much  facilitate  the  labors  of  parents  and  teachers. 
Yours  truly, 

J.  H.  Brown,  A.  M. 
Prin.  of  Zane  St.  Boys'  Grammar  School 

From  Professor  James  C.  Booth,  A.  M,,  M.  A.,  P.  S.  Author  of  the  Ency- 
clopedia of  Chemistry;  Melter  and  Refiner  in  the  U.  S.  Mint;  Pro- 
fessor of  Applied  Chemistry  in  the  Franklin  Institute. 

Analytical  Laboratory,  College  Avenue,  ) 
Philadelphia,  July  3, 1851.  { 
Dear  Sir — I  have  examined  "Familiar  Science"  with  some  care, 
and  must  express  a  hearty  approval  of  the  manner  in  which  the  most 
"common  things"  of  life  are  familiarly  and  clearly  explained, 
without  sacrificing  the  correctness  of  science.  Embracing  such 
questions  as  are  usually  put  by  the  developing  mind  of  children,  with 
clear  and  precise  answers,  it  will  relieve  parents  and  teachers  of 
the  unhappy  necessity  of  crushing  youthful  inquiry,  while  it  will 
tend  to  nourish  a  spirit  of  reflection  and  investigation  in  young  and 
old.  I  commend  it  as  a  valuable  catechism  for  schools,  and  for 
amusement  and  instruction  at  the  fire-side. 

Respectfully  yours,  Jas.  C.  Booth. 
Me.  R.  E.  Peterson,  Phila. 


From  Mrs.  Phelps,  favorably  known  as  the  author  of  various  popular 
Bchool  books  on  Chemistry,  Botany,  Philosophy,  Geology,  etc.  etc. 

Mr.  Peterson. — Sir — In  thanking  you  for  the  book  called 
"Familiar  Science,"  permit  me  to  express  the  satisfaction  with 
which  it  is  received  by  myself  and  ihe  teachers  and  pupils  of  this 
Institution ;  having-  labored  for  many  years  to  render  Science 
Familiar,  I  am  glad  to  meet  with  this  unpretending  assistant.  It  is 
recommended  by  the  fact  that  if  adopted  in  a  school,  it  will  crowd 
out  no  other  book,  since  it  is  sui  generis  and  may  be  studied  as  an 
amusement. 

With  respect,  yours,  etc. 
Almira  Lincoln  Phelps, 
Patapsco  Institute,  near  Ellicott's  Mills ,  Md. 
June  30, 1851.  

From  the  Rev."  Father  Sourm. 

Philadelphia,  July  4,  1851.     ) 
St.  John's  Cathedral.  ) 
Dear  Sir — I  comply  most  willingly  with  your  request,  though  after 
the  high  and  well-deserved  praise  which  your  valuable  work  has  al- 
ready received,  further  commendation  would  appear  unnecessary. 
An  examination  of  its  contents  will  "satisfy  every  intelligent  reader 
that  it  is  a  treasure  of  various,  useful,  and  entertaining  knowledge, 
well  calculated  to  promote  one  of  the  chief  purposes  of   true 
education,  viz.  to  make  us  more  familar  with  those  works  of  God 
''so  often  full  of  singular  beauty  and  power  When,  seemingly,  most 
common)  in  the  midst  of  which  "  we  live  and  move,  and  have  our 
being."  It  will  afford  me  much  pleasure  to  aid  in  its  circulation. 
Kespectfully,  your  obed't  serv., 

Edw.  J.  Sourin. 
fl.  E.  Peterson,  Esq. 


From  T.  S.  Arthur,  Editor  of  the  Home  Gazette. 

"Familiar  Science,  or  the  Scientific  Explanation  of  Common 
Things,"  edited  and  published  by  Robert  E.  Peterson,  of  this  city,  is 
one  of  the  most  generally  useful  books  that  has  lately  been  printed. 
This  work,  or  a  portion  of  it,  came  first  from  the  pen  of  the  Rev. 
Doctor  Brewer,  of  Trinity  Hall,  Cambridge,  but,  in  the  form  it  first 
appeared  from  the  English  press,  it  was  not  only  unsuited  to  the 
American  pupil,  but  very  deficient  in  arrangement.  These  defects, 
the  editor  has  sought  to  remedy.  To  give  not  only  to  the  parent 
a  ready  means  of  answering  inquiries,  but  to  provide  a  good  book 
for  schools,  is  the  object  of  this  volume.  About  two  thousand  ques- 
tions, on  all  subjects  of  general  information,  are  answered  in  lan- 
guage so  plain  that  all  may  understand  it.  We  had  marked  a  num- 
ber of  these  for  publication,  but  must  defer  their  insertion  to  another 
number. 

The  book  is  very  strongly  recommended,  and  not  without  good 


reason,  by  competent  judges  of  its  merits,  and  we  add  our  own  tes 
timony  in  its  favor. 


From  the  Episcopal  Recorder. 
We  are  here  furnished  with  one  of  those  time  and  labor-saving 
volumes  which,  by  judicious  arrangement,  simple  illustration,  and 
accurate  definition,  greatly  facilitate  the  important  effort  to  impart 
a  familiar  knowledge  of  the  facts  and  principles  of  natural  science. 
To  all  interested  in  the  education  of  youth,  it  will  be  found  a  valu- 
able auxiliary,  while  its  study,  in  families  and  schools,  cannot  fail 
to  enlarge  the  domain  of  Science,  by  fixing  in  the  mind  the  elemen- 
tary principleis  of  the  every  day  phenomena  of  life. 


From  the  Saturday  Gazette. 
One  of  the  most  valuable  "family  library"  books  we  have  ever 
met.  The  title  explains  the  object  of  the  work,  which  embraces  a 
wonderful  variety  of  topics  cleverly  treated.  Twenty-five  thou- 
sand copies  of  the  English  edition  v/ere  sold  in  two  years,  and  the 
present  publisher,  also  the  editor,  has  made  it  worthy  of  as  large  a 
circulation,  by  the  clear  type  and  paper  and  the  durable  binding. 
We  commend  it  to  all  parents  or  teachers  of  the  young. 


From  Fitzgerald's  City  Item. 
Occasionally  a  book  is  published  that  fills  a  void  that  seems  to 
have  been  left  for  it — that  exactly  furnishes  what  has  been  ofteu 
wished  for.  Such  a  work  is  "Familiar  Science."  It  gives  con- 
cisely, yet  clearly,  about  two  thousand  explanations  of  the  scientific 
principles  involved  in  the  most  frequently  recurring  operations  of 
life.  It  is  an  invaluable  book  for  all  whose  relation  with  the  young, 
whether  as  parent  or  teacher,  demand  a  facility  in  explaining  what 
so  often  attracts  the  curiosity  of  childhood.  To  the  general  reader, 
too,  it  opens  an  interesting  field  for  observation ;  and  even  to  the 
scholar  accustomed  to  philosophical  investigation,  it  may  present 
familiar  applications  of  what  are  often  regarded  as  abstruse  and  un- 
practical researches. 


From  the  Saturday  Evening  Post. 
Here  is  a  book  which  might  find  an  appropriate  place  in  every 
school,  and  in  every  house  where  there  are  young  inquiring  minds, 
athirst  for  knowledge.  It  contains  nearly  two  thousand  questions, 
with  scientific  answers,  simply  arranged  and  clearly  expressed,  con- 
veying, in  a  small  compass,  a  great  amount  of  information  respect- 
ing the  every-day  wonders  which  puzzle  the  restless  brains  of 
children,  and  sometimes  their  elders,  too.  The  publisher  and  editor 
merits,  and  will  no  doubt  receive,  the  thanks  of  all  parents  and 


teachers  who  have  experienced  (and  what  instructor  of  youth  has 
not?)  the  difficulty  of  meeting,  with  satisfactory  explanations,  the 
innumerable  questions  of  those  under  their  charge.  The  book 
needs  only  to  be  known,  to  be  pronounced  by  a  discerning  public  a 
rich  fund  of  useful  entertainment  for  the  household  circle,  as  well  as 
a  valuable  aid  to  instruction  in  schools.  We  are  not  surprised  at 
the  statement  in  the  preface,  that  "Twenty-five  thousand  copies  of 
the  English  edition  of  this  work  were  sold  in  London  in  less  than 
two  years;"  neither  shall  we  be  astonished  if  double  that  number 
are  sold  in  the  United  States  in  even  a  shorter  space  of  time. 


From  the  Banner  of  the  Cross. 
It  is  an  exceedingly  interesting,  as  well  as  instructive  book  for 
children.  It  teaches  them  the  reasons  of  many  things,  which 
though  common,  are  yet  mysterious;  and  it  also  gives  many  practi- 
cal hints  upon  matters  which  relate  to  the  experience  of  every-day 
life.  *  We  can  heartily  unite  with  the  many  strong  recommendations 
from  those  whose  opinion  in  such  matters  are  especially  valuable, 
in  their  favorable  judgment  of  this  work,  as  a  most  excellent  book 
for  school  or  family  use. 


From  the  Christian  Observer. 
This  is  an  excellent  and  valuable  book  for  the  young  in  families 
and  schools.  It  opens  to  the  reader  a  rich  repository  of  science  re- 
specting the  most  common  things — facts  and  lucid  explanations  of 
them  which  cannot  fail  to  interest  and  instruct  those  for  whom  it  is 
intended.  We  commend  it  to  the  public  as  a  valuable  book  for  the 
home  library  and  the  school. 


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